Prussia joins Austria-Hungary and beats Napoleon III in Italy and France.

OTL 
  Prussia allowed Napoleon III to trounce A-H in Italy, thus depriving the German League of this valuable territory and enabling the unification of Italy. Then Bismarck and the Kaiser made the huge mistake of inducing the GL to exclude A-H, so that when Germany was unified after defeating France (which the GL had to do without A-H's assistance), it did not include A-H. A-H had a stronger force than France in Italy (although Napoleon send half his army to Italy quite fast, using trains for the first time for effective transportation of large forces), with 22,000 cavalry and 220,000 men, but they were poorly lead and promptly defeated.

ATL
  Prusia avails itself of the fact that Napoleon has half his army in Italy, the perfect occassion to defeat France with A-H's help and to save Italy for the GL. Accordingly, when A-H is defeated in Monebello on May 20, 1859, Bismarck and the Kaiser send 10 divisions to Italy to reinforce A-H and defeat the French expedition. Prusia also convinces all members of the GL (Bavaria, Saxony, etc,) to invade France from the east with the rest of the Prussian army, catching Napoleon III completely out of balance and forcing him to fight in 2 distant fronts, separated by mountanous, neutral Switzerland and against mush stronger forces on both fronts.

  The completely unexpected GL attack from the east causes Napoleon III in Italy to panick, even before the Prussian divisions engage him in Italy. The GL advances rapidly to capture Sedan and Reims, causing rapid withdrawal of the weak French forces all over the front to defend Paris. The Germans chase them closely, capturing most of the guns, munitions and supplies.

  When 10 aggressive Prussian generals arrive in Italy, they immediately deploy the massive cavalry to cut off Napoleon's supply lines and possible routes of retreat and attack Napoleon's force in a pincer movement, which smashes the French van, rapidly defeating and capturing Napoleon III, the French, Piedmonetese and Sardinian fieldmasrshals and generals.

  The fall of Napoleon III renders the defense of Paris moot against much stronger forces, so French forces capitulate everywhere, including the colonies, which cannot endure withtout support from France.

  A-H not only keeps Italy, but the massive Prussian-AH force in Italy and the GL forces in France, Bavaria and Austria immediately invade Switzerland, causing its capitulation in weeks. GL forces in Gallizia, Hungary, etc, invade Romania simultaneously, while GL leaders convene in Versaille to unify Germany (including A-H and its Italy and annex France and its colonies, Switzerland and Romania. Prussia also rapidly invades weak Denmark without declaring war and incorporates its navy and commertial fleet. Germany builds ships in French and Danish shipyards, rapidly expanding its navy and commercial fleet (already boosted by France's and Denmark's)

  Massive Germany then presents Holland with an offer to request annexation by Germany, which will send a million settlers (including many times more Dutch than have migrated in centuries, because they will have jobs and modern towns and cities to live in) to secure and rapidly develop the barely settled (a few thousand Dutch) DEI, Suriname and A,B, C Islands, build railroads, roads, shipyards, steel, flour and sugarmills, lumberyards, factories (nutritious marmelades of tamarind, guava, pineapple, blackberry, strawberry, etc, made with healthy brown sugar and amaranth, clothing, munitions, bicycles, etc, fish, fruit and meat canneries), universities, schools, hospitals, etc, in dozens of new cities and hundreds of new towns and to clear the jungle for spieces (cinnamon, cloves, nutmeg, etc,), cashews, peacons, almonds, coffee, cacao, rubber, tamarind, coconut, banana, guava, avocado, mango, licorice plantations, grasslands, sugar, rice, tobacco, amaranth, peanut, ginger, cabbage, tomato, strawberry, blackberry, pineapple, turmeric, hyabiscus, potato, oatmeal, soy, lentil, bean farms, cattle, sheep, fowl, shrimp, crab, oyster and pig farms and ranches. To exploit efficiently in a large scale fishery, mining, logging, tourism, etc,  
  Germany will also provide work around the clock in Dutch shipyards if Holland is annexed by Germany and steel to build them. Unemployed Dutch will be able to find jobs anywhere in germany, including Dutch and French colonies.
  Otherwise, Germany will invade the DEI with the former French navy and the huge German army will invade Holland rapidly after much useless bloodshed. Holland requests annexation, providing its navy, foreign trade institution, excellent shipyards, textile and shocolate fctories, an enterprising, well educated population, etc,

  Deploying its navy (now including the French, Danish and Dutch navies) and 8 divisions of excellent infantry and cavalry (now including French, Dutch, Swiss troops) in 1861 Germany invades weak Afrikaner South Africa, which has under 100,000 people. Germany sends 200,000 settlers to develop and secure the colony.

   In January 1862 Germany persuades Britain to use Canada, Jamaica, British and German Guianas, the Falklands, etc, and  the formidable German and British navies and armies to invade and share the invaluable northern US and Latin America, while the US is heavily engaged in the civil war and all Latin America has small and weak navies and armies. Britain will recover the former American colonies and acquire large, valuable territories in Latin America at a small cost, compared to wars in Europe. Britain agrees and within months a large expeditionary force disembarks in Atlantic city, without declaring war.