Until you become a British citizen (British passport, etc), you're still looking at going through the non-EU nationals line (vs the EU nationals line). The fiance visa, spousal visa (LLR or ILR), etc -- you're still an American national with an American passport, not a national of a European Union country.
TBH I always have observed that the British line (EU line) is longer than the non-, but I fly in through Manchester. Maybe the lines shape up differently at other airports.