WI: US stays part of Britain. What does the government of the 13 colonies look like in the future?

That would be a fun timeline to plot out.  You could have a continent with multiple dominions grouping the colonies into cultural regions to keep them more peacefully coexisting.

1. Multiple dominions:
-Dominion of New England (Mass, Rhode Isl., Conn., Vermont, New Hampshire, Maine, Nova Scotia) with 10 provinces (Nova Scotia being split into Nova Scotia, New Brunswick, Cape Breton Island, and St. John Island, and Maine separating from Massachusetts); capital at Boston
-Dominion of North America: 
--bounded by the Ohio River and the Mississippi (US 1783 border), due north to Hudson Bay (later adjusted to the Nelson River flowing into Hudson Bay)
--provinces include New York, New Jersey, Pennsylvania
--Ottawa province (Ontario peninsula bound by the French River, Lake Nipissing, Mattawa River, and Ottawa River)
--Illinoia Province (Illinois, but eastern border follows Wabash River until the Tippecanoe River splits east/west, then due north to the Great Lakes, just east of our Long Beach, Indiana)
--Ohio Province (Maumee River west to the St Marys, then due south until it meets the line due east from the Wabash)
--Michigan Province (OTL Michigan plus the portion of Indiana not in Ohio and Illinoia)
--Wellington Province (Wisconsin, Michigan's Upper Peninsula, and the portion of Minnesota included south of the Rainy River), capital is Carleton (OTL Madison)
--Mississagi Province (Ontario north of this Ottawa Province, until the 49th parallel and the Great Lakes)
--West Hudson territory (remainder of Ontario to the Hudson Bay)
--Quebec (OTL Quebec north of the St Lawrence, to the 49th parallel and east to Saguenay)
--Montérégie (OTL Quebec south of the St Lawrence)
--East Quebec (OTL Quebec east of the Saguenay River and south of the 49th parallel)
--East Hudson territory (northern Quebec)
--total of 12 provinces, 2 territories
-Dominion of Newfoundland (same as OTL Newfoundland and Labrador)
-Confederation of Virginia
--Virginia, North Carolina, South Carolina, Georgia, East Florida, West Florida, Mississippi (WF and MS are equally split north/south), Kentucky, Tennessee, Bahamas, Cuba, Puerto Rico, Santo Domingo, Virgin Islands, Maryland, Delaware, Bermuda
--When UK abolished slavery, they got a compensated emancipation, as they had over a hundred abolition societies (same as OTL), sent some freedmen to Liberia and Sierra Leone.
-Dominion of Columbia (Oregon Country plus remainder of Canada and Alaska, east of this Dominion of North America)
--Provinces: Alaska, Vancouver Island, Oregon (OTL plus little bump due to Snake River), Shoshone (what we'd call Idaho), Victoria (OTL Washington), Columbia (OTL Columbia to the 52nd parallel), Prince George (OTL BC north of the 52nd), Yukon (OTL Yukon and Northwest Territories to the 120 W); from 114 W to 93 W, three provinces 6 degrees tall, Alberta, Saskatchewan, Manitoba.  Two territories - North Hudson (56 N to 61 N), Arctic Territory (120W to the east, 61N to the Arctic Ocean)
--11 provinces, 2 territories
-Republic of Texas (OTL Texas, Rio Grande, Alta California, Baja California, New Mexico, Sonora, Chihuahua, Durango, Sinaloua, Yucatan)
--more similar to what we'd consider American.
--13 states: Rio Grande, Yucatan, Sinaloa, Durango, Sonora, Chihuahua, New Mexico (bordered by the Rio Grande river to the west, 100 W in the east, 32 N south, Arkansas River to the north), Colorado (42 N to 38 N, 106 W to 113 W), Arizona (38 N to 32 N, 113W to the Rio Grande, but with the coastline from the colorado river to the Sonoyta River), Baja California (Baja peninsula to the 38th parallel, west of the mountains) North California (38th parallel to 42nd, west of the mountains); North and South Nevada
-Dominion of Louisiana: Louisiana purchase territory
--OTL Louisiana Purchase with OTL borders with Texas
--14 provinces, French/English are languages, New Orleans is capital.
--Flag is same as Quebec but golden fleur-de-lis.
Flags:
 (New England)
 (Dominion of Columbia)
 (Flag of Louisiana)
 (Dominion of Virginia, using Scottish and Irish flags with stars for each province)
 (Texan Flag, one star for each state)
 (Dominion of North America flag, but with 12 stripes, not 13)

Each dominion (DNA, D Col, D Vir) would have a Governor-General representing the crown, a Prime Minister and Cabinet, then a Congress, most likely, with a Senate and a House of Delegates.  They'd likely have a right to bear arms due to the colonial heritage, but without a war with Britain to solidify the other rights, those are all up in the air, including the right to freedom of religion.  Texas would likely be the freest country, filled with refugees seeking free speech, freedom of religion, and the rest.  The Virginia Dominion would be populated mostly by Scots-Irish, some Germans, including Hanoverians, and Afro-Virginians, giving it the St George's Cross with the St Patrick's cross within it.

Virginia would to this day be a patriarchal society, valuing family and tradition most, and resisting rapid change for the sake of change, and wanting as little government as possible; New England, in contrast, would have a very activist government, and would have enough regulation that businesses have a tough time staying profitable, like California.  Dominion of North America would be a middle ground.  New England has the pilgrim tradition of busybodies telling the other dominions what to do, but without a political union with the other dominions, they can ignore them safely.  The D of NA has a culture more similar to that of the Dominion of Virginia without New Englander immigration influencing them.

New England had full prohibition for about 20 years, but it was repealed in the 1930s.  Its politics are very left-leaning, with its right-most mainstream political parties similar to those in Europe and Britain, about how we imagine a 'RINO' Republican.  Most politicians support progressive income taxes, the 'Bank of New England,' government-funded health care and social security and state income taxes.  The Labour Party is similar to the 90s US Democrats, while the Conservatives favor fewer tax brackets, more business-friendly regulations, and partial privatization of some government welfare to make it less wasteful.  New England has meddled in foreign countries' elections to help its profits, industrialized early on, and had an early environmental movement.  Despite its moralizing, New England is the least tolerant of others, be it people of other races or political viewpoints.  This country had plenty of immigration of the radicalized 48'ers from Germany, a bunch of marxists, who also went into the Dominion of North America, influencing their politics as well.

Dom of N.A. is kind of in a middle ground politically.  Its culture is less radical than New England but has a very moderate socialist tendency.  Its welfare system is a flat 5% tax paid into a private account invested in gov't approved funds, plus your own private contributions through your employers.  The first two years of contributions are partitioned into your own unemployment account, and the provinces have their own Medishare programs.  Industrialization proceeded similarly to OTL, with tariffs funding internal improvements until the late 19th century when small income taxes began to be passed.  Progressivism in this dominion brought about industrial safety legislation, clean air/food laws, and some limits on politicians' terms of office.  The population is mostly English, though some Celts do immigrate here.

Dom of Virginia is the most culturally conservative, but is more racially tolerant, having had no Tariff controversy, no struggle with New England over slavery, no Civil War, no Reconstruction, but found its own way to live with the minorities and majority-minority counties.  Emancipation caused slavery to be replaced over a ten year period with indenture, a period of adjustment to freedom for the freedmen.  This dominion got a lot of immigration from Wales, Scotland, and Ireland of the more conservative elements of those societies, causing Irish, Scots, Scots Gaelic, and Welsh islands in these provinces, as well as plenty of Germans seeking escape from the restrictions of the German Empire.  Music would resemble jazz, blues, rock, folk, with songs having hurdy-gurdy, violin, guitar, and drums.

Without a 'United States of America' a number of events would change.  WW1 might still happen; if you real 'Killing of Uncle Sam,' it was more about the United Kingdom maintaining its power and influence over Germany, so they would find an excuse to attack, but with multiple dominions still part of the empire, it's possible the British could get them into the war without a Lusitania false flag.  But if the war ends sooner, dominion populations might urge the UK not to go scorched earth on Germany, like France might want, leaving a Germany without Alsace-Lorraine, a smidge of Silesia, and some of Posen, and otherwise intact, and keeping its monarchy, just without Wilhelm II at the helm, perhaps Prince Henry; Austria-Hungary would federate as planned, but then peacefully break up into ethnic states.  There is no global depression because there is no Federal Reserve monkeying with credit and currency for the largest economy in the world.  At most, maybe New England experiences a depression, which would affect other local dominions trading with it, but they would pull out of it sooner than New England.  There is no WW2 unless it's engineered, as this Germany doesn't have a war guilt clause or hyperinflation or unstable republic to help provide a place for radical parties to grow.  Fascism does grow in Italy, France, Spain, and elsewhere, and maybe France would start a war with Germany, perhaps with Poland providing a distraction in the east, but it would not grow into a world war.  At the least it would end within a few short years or even months, with either a status-quo, a linguistic-split Alsace-Lorraine, or at best for Germany, a restoration of its earlier borders and cancellation of its WW1 debts to France, and a population swap with Poland, regaining Posen and its lost bit of Silesia.  Poland would end up being the small Congress Poland plus Austrian Galicia.

Russia's Bolsheviks were funded to the tune of $20 million by Jacob Schiff of New York's Kuhn, Loeb and Company, which would not happen this timeline, giving the Russian Czar a good chance to survive.  If WW1 is shorter, Russia would suffer less, but its finances and manpower loss would help push for some more liberalization and modernization, within the monarchical framework of Russia, giving the Duma more power, despite the czar's wishes.  Without the US helping fund Soviet communists, China's nationalists might not have as hard a time against their communists who would likely not have czarist backing, leaving a nationalist China that would slowly open up and liberalize like Taiwan did OTL.

There is no CIA in this timeline, so Iran's Mohammed Pahlavi might still be on the throne past 1953.  If France in this timeline keeps its Indochina colonies, there is no Vietnam War with the US, just a French and maybe British-guided decolonization.  Africa, if Germany's colonies aren't taken after WW1, would still experience decolonization, maybe a decade or so later, though.  It would be unsure how it would proceed, and if there would be any white-settler-majority, perhaps Namibia, maybe Rhodesia or South Africa, depending on circumstances since the 18th century are different.  Without WW2, fascism as an ideology might die out or milden to just another form of socialism, and the space race, which was aided by German rocket scientists secreted into the US, would be a later event, if it happened, but would not be fueled by a US vs USSR antagonism.  If it happens, it would likely be between British Dominions and Texas, where the best rocket launch sites are.

Just a small thought.

In other words, no reason for any hypothetical free soilers to side with the British. They might argue with the Slaveholders about what to do with the land, but that is secondary matter to the immediate issue of the British not letting them settle the land at all or at least at a relevant timeframe.

I didn't argue that, however. I have argued that, with the continuation of the slave trade due to the Crown, slavery as an institution would be much more widespread than OTL, reducing any free soiler political movement. Ignoring any disagreements on that particular point to focus on the big picture, any Free Soil movement has no reason to side with the British at all; they may disagree with the slave holders how to use the land, but that's irrelevant until they are allowed to settle said land anyway. You're making the assumption they will hate the slave holders more than the British despite neither having access to the land in the first place to fight over. 

Likewise, I have a very poignant question for you: 75% or so of the Confederate Army IOTL didn't own slaves and there was no real Anti-Slavery movement in the South for the most part after 1830. Why would there be a massive Free Soil movement here in an expanded slavery scenario when there wasn't one in the South itself IOTL?