it's hard to see what options they would have had if the invasion had failed catastrophically or the Invincible been sunk.
go back to the Isles, build another fleet, come back in 3-5 years? what other options did they have? go to Paris and beg them for the Foch & the Clemenceau? After their own aircraft carrier had been destroyed? I would have loved to have sold Thatcher the insurance policy...
HMS
Illustrious was completing fitting out when the Falklands War started and IRL she arrived in the islands in late August. If either
Hermes or
Invincible had been knocked out (sunk or unable to operate aircraft) prior to the landings at San Carlos, then I think the Royal Navy would have pulled back the surface fleet, left the Argentine troops to shiver over the winter (with subs making resupply difficult), and then returned with
Illustrious and the surviving aircraft carrier in the spring.
If a carrier had been lost after the landing, then isn't it plausible that the British would just have carried on pretty much as in real life? A primitive VTOL airstrip (good enough for Harriers) was built at San Carlos within a few days of the landing, so they would still have had two operating bases. It really depends how many Harriers were left.
If
both carriers had been lost, it would thave been possible to wait until
Illustrious and
Ark Royal were ready; even with an emergency crash fitting out this would have been difficult to complete before the (Southern Hemisphere) winter of 1983, more realistically the autumn of 1984 (two years after the invasion). Unless the Argentine aviators had been incredibly, incredibly lucky, this would presumably have meant that a large number of destroyers and frigates had been hit as well. The Royal Navy sent just under half its escorts in the CORPORATE Task Force. If there had been more losses, it would certainly have been possible to abandon all their NATO commitments, pull ships out of refit, and replace the losses for another task force in 1983 or 1984. The most difficult part would have been aircraft losses, but the Sea Harrier production line was in full swing. IRL the average output was about 6 planes a year, but maybe they would have been able to get a basic carrier squadron (12) ready in 18 months or so.
But the real problem would have been political. Would there still have been support for recapturing the islands after taking such devastating losses (e.g. 2 carriers + 4 destroyers + 4 frigates + a troopship... that's an awful lot of corpses)? A returning Task Force would have faced even higher odds, because Argentina would surely have built a fast jet airfield on the Falklands, given enough time. TBH, I think the answer is: yes. Thatcher would have had to resign, but the 1940 myth ('Britain fights alone') is so foundational to late 20th century British identity that I think the political will would have been there to try again.
If
four carriers had been sunk, then I agree that it would have been game over!