Technically the Channel Islands and the Isle of Man are semi-independent (as they are directly administered by the Crown, with their own governments, and hence not part of the UK), which I would think means they are not subject to the First Lord and after. So the normal machinations of government would probably go ahead as if nothing happened (even when submitting laws to the Privy Council for approval). The other Overseas Territories, like Bermuda or Gibraltar, OTOH, would be something I'd like to see more of from @powerab - we already have it as canon that HK is lost and a *Falklands War does happen, though the international response is different from TTL in both cases and only serve to highlight the junta's pariah status. (The Isle of Wight, OTOH, is unfortunate in that as part of England it is subject to everything happening so far ITTL.)Probably kept the overseas territories like the Channel Islands, Isle of Mann/wight
So with the junta gone, I would assume a newly democratic UK would be in a better shape to negotiate with equally-democratic Argentina and Spain over the Falklands (plus SGSSI) and Gibraltar, respectively. Eventually, they would probably come up with some sort of special status for both (Spain already has such things embedded in its democratic system, while Argentina would just have to reactivate the National Territory status, unused since 1990 with the conversion of Tierra del Fuego into a province, and use it as a base for creating an autonomous region of sorts), maybe even cases of transitional joint sovereignty in the case of Gibraltar especially. So that could have also been another trigger for the British 23F - much like negotiations for autonomy IOTL with regards to Galicia, Catalonia, and the Basque Country were a big factor in Spain's own 23F.