The real question is, as you've alluded to, is will the UK government try again. Certainly the Thatcher government is finished, but the question is what happens next? A reshuffled Conservative government? An immediate election? Who'd win such an election? Is there a hung parliament, a small Labour majority under Foot or a small Conservative majority? Does the Lib/SDP block win many seats?
I doubt there would be an immediate election, because the Tories would probably be unable to win it, and for the moment there majority is solid enough to keep going. The only possibility for an early election is if there is a mass defection to the Alliance, and then the government loses a vote of confidence. But a new leader would make that more unlikely, especially if it is a wet like Whitelaw. Maybe if Thatcher somehow stayed on, but other than that, the government would go on, probably till the end of their term.
The Alliance might retake the lead in the polls once more, and win a few by election to keep that going, but ultimately, with a new PM, and a recovering economy, they might well lose steam in the following two years, and because of FPTP, they need to not only beat Labour and the Tories in the popular vote, but do so by a significant margin if they hope to emerge as larger than either. If it is a tight run thing, with them scoring in the low to mid thirties, the most likely result is that they back a continued Conservative government in exchange for PR.
UK becomes laughing stock. Short and simple. The whole idea "From Empire to small country" would get instant shot in arm.
Expect more multicultural UK, with more immigrants, stronger new Labour party and forget about leaving EU.
I would expect in every debate in "should UK have referendum for leaving EU" the point " We lost Falkland war we must be in EU since we are weak".
There would be no EU referendum. More colonies would be lost. Whole handling of Honk Kong would be more pro Chinese think.
T.L.D.R: UK would more multicultural and leftist.
Labour were actually advocating leaving the EU at this point. But in any case, they would not be strong enough to form a government, given how divided they were at this stage. Maybe they would end up effectively fluking the most seats due to the vagueries of FPTP, but even in that scenario, the SDP and the Liberals would likely join with the new Tory PM in coalition to lock them out of power. The Tories were more vulnerable to the Alliance at this point than Labour. Had the Falklands been lost, they would have shed votes to them, the question is if it would be enough to get them to lead a government.
Not sure we would be more leftist, at least not in the short term - coming round to the arguments that the UK would get more militaristic and dick-waving to compensate. We'd want people to know they definitely shouldn't mess with us, we're still a big deal, look at our stuff etc etc. SDP-Liberals might win the election but they'd do it still promising 'strength', Labour will be left in the cold until they do the same.
(This is not going to be fun for Northern Ireland)
An interesting question is whether it would swing the SDP leadership election, which took place shortly after. Owen raised his profile considerably in that campaign as the SDP's spokesman in parliament, and he looked a good deal more patriotic than any of the Labour or other Alliance leaders. If a loss in the Falklands had swung the contest for him, then the Alliance has a leader who is much better suited to populism than Jenkins, and can appeal to Tory voters quite well too.