I take it you mean the Argentinian bit of TdF. I've never had any problems in Ushuaia with a British flag boat or British passports although a few years ago the Aduana ( customs) were giving all the charter
boats a hard time regardless of flag...just because they could.
The only issue would be if you were coming from the Atlantic side and stopped off at the Falklands without getting 'permission' from the mad cow in the Casa Rosada. That would cost you big $$$$ if you then went to Ushuaia. ( edit..ooops...see you are caribbean bound so in that case if calling at the Falklands just clear from PW (Chile) and don't go back to Argentina after the FI.)
Of course you can go from the FI to Puerto Williams and not call at Ushuaia...it is possible to store and fuel
and everything at PW ( you can get stores shipped down from Punta Arenas on the weekly ferry
if stuff you want isn't available locally).