Possibly, but the UK governments during the era when the Empire was dispersed and sent off to go its own way was never very happy with the idea of integrating parts of the Empire into the UK proper. The closest it came was when the Maltese wanted to be integrated and were offered seats in Parliament, but a diplomatic incident created a lot of ill-will towards the British government in Malta and they pulled out. After that, there was never really any enthusiasm for uniting the colonies with Mother Britain. Rather, the UK - depending on your take on history, either deciding that it needed to get rid of the colonies for economic reasons, or feeling betrayed by the American stance on decolonialism and thus feeling like it was "obliged" to give colonies independence whether they wanted it/were ready for it or not - ditched any colony or dominion which was self-sufficient. In this case, depending on the strength of Tierra del Fuego it has two options really. If it's in a position where it could support itself then it will be cast away and told to fend for itself - in this case it might be given the Falklands as a goodwill token unless the UK decides it needs the naval base. Otherwise, it will be kept on in the role of crown colony, the same as places such as Gibraltar and the Falklands OTL. It's possible in this scenario that TdF could eventually become strong enough to request an independence ballot, but in this scenario you'd probably still get the Westminster Parliament saying that it wasn't willing to offer full union, only colonial status or independence.
To have a TL where TdF becomes a full-fledged part of the UK you'd probably need to alter events so that the British were much happier to integrate colonies, which probably requires a much softer take on decolonialism by the Americans when the analogue of the post-war years comes around (i.e. when the realisation that it's time for the Empire to be severed comes). Of course, with this change it's possible that you wouldn't just see one big dumping of the colonies. With a "soft decolonialism" you could see the big and relatively affluent colonies the likes of Egypt, Singapore, South Africa, Jamaica etc etc be given their independence in their own time, and the very poorest colonies, with the weakest political systems and also the places most likely to fall into civil war being kept on by the UK in a sort of "we promise we will get your economy working before we let you go" idea - or at any rate, since the African colonies were a huge money-drain, you could see the UK holding onto a few problem colonies such as the future Zimbabwe, Sierra Leone etc, and those colonies could feasibly - if native African ill-will towards Britain calmed down and the white ruling classes were persuaded somehow to share power with the natives, and if their economies still failed to really properly start up, you could see an interesting thing where these colonies become fully integrated into the UK in a kind of "you missed the independence boat, now times have changed and no-one wants to separate anymore" thing.
Well...that last paragraph really had nothing to do with TdF. I got a bit carried away, sorry. I'm a bit of an Empire apologist at heart. But anyway, hopefully that gave you some idea of what I was trying to say, anyway...
