Can Callaghan hold on?

  I think his big mistake was not going to the polls in Oct 78, when he was clearly ahead and most people were expecting him to.  His famous "waiting at the church" joke when a press crowd expecting the announcement of an election were told he wasn't going to call one then.

  After this we had the collapse of his wage restraint programme and the "Winter of Discontent" and this let Thatcher and the Tories into power and the rest is history.  If he had gone then and won a clear majority you would almost certainly had union problems during the winter but there is a chance that the Labour party, still largely dominated by its right wing elements, would have gone for forced reforms to restrain the unions rather than Thatcher's desire for total reversal with [mis-]management having largely untrammeled power.  Also it would have been Labour rather than the Tories who would have benefited from North Sea Oil rather than the Tories and it would probably have been used better - since its difficult to see it being wasted worse that Thatcher did with pushing unemployment so high along with massive tax cuts for the very rich and asset striping.  Plus they would have had a business policy so while probably managed poorly it wouldn't have dumped leading areas of the economy, such as computing and alternative energy like the Tories did.   Would very likely still have had a lot of wastage but again I doubt they would have mis-managed the economy as badly as the Tories.

  Best option would have been a basic level of reform and continued development then possibly a loss to a Tory party in ~1982 which would have been reformed, having dumped Thatcher's ideological crap as a disaster.  Both parties would have continued to be far from optimum but a lot better than the government we're had since 79 OTL.

  Anyway diversion over.  In terms of the OP I doubt him clinging on to October.  His government were fatally wounded by the chaos the unions made of the country during the winter.  Especially since nobody realised at the time things could get even worse.  Most likely he would still have gone in May or June as I doubt he could maintain support much longer plus hanging on until he had to go would have look like a broken government clinging to power and probably meant he lost even more heavily.

  Very true.  Possibly similarly if May had gone for a mandate in 2015 when she came into office instead of saying she wouldn't then doing so a year later.

  This assumes that Labour is still in power when it happens.  IIRC there were reports that intelligence suggested the Argentinians were looking at an attack ~1978 and the government sent a SSN to the S Atlantic and let the junta knows that and the latter decided it wouldn't be a wise move to pick a fight with Britain.  Might be that presuming they get the same warning they respond on a similar way, which might make a more desperate junta back down or might not.  If they don't get warning or ignore them then the war would be on pretty much as OTL although a lot would depend on how the government had gone in the previous few years.  Pretty certain a Labour government would fight but how the forces available to it varied from OTL I wouldn't know.  In theory a Labour government is more likely to make cuts but Thatcher had announced cuts in naval spending, such as selling off one of the new Invincible class CVs and scrapping the old ice patrol ship Endurance, which was operating in the S Atlantic and has been seen as what persuaded the junta that Britain wouldn't fight to retake the islands.  Fortunately they were wrong and also we managed to win.

  Not sure if the intelligence services didn't pick up on hints in 82 or whether they did and the Tory government ignored them.

  Probably basically in a similar way.  Instruct the military to liberate the islands and largely leave them to do it.  Might be a little better handing of intel as there were some reports that planned attacks, think mainly leading up to Goose Green were leaked by the government for publicity reasons, which seems bloody stupid, even for Thatcher and her cronies.

  The key factors might be any differences in force levels and preparations and if say a Labour government in Britain, say having won an election in autumn 78, then what has happened in the US and relations with whoever is President there.