It's possible you heard about the Scottish or Welsh parliaments within the UK (like a state government in the US, although with somewhat more power I believe), which partially work like that. Or you've just gotten us mixed up with one of our European neighbours that does do it. Sweden and the Netherlands use a system like you described.

The UK's House of Commons, which I'll refer to as Westminster going forward, is our equivalent to the American House of Representatives. We've also got the House of Lords, which is our equivalent to the Senate, but it's unelected (largely chosen by each outgoing prime minister) and far, far less powerful than the US Senate. It can't make Westminster pass or not pass something. Anyway, Westminster is elected by first-past-the-post. 650 constituencies, each one is considered totally separately from the other, highest number of votes for a candidate in that constituency gets the seat. Whichever party has the most seats gets to try to form a government first, either with its own majority, in coalition with a minor party if it doesn't have one (happened recently with the Conservatives and the Northern Irish party the DUP), or just as a minority government if the opposition is unable to form a larger coalition.

Situations like you describe where A and B try to win the allegiance of C do happen, particularly when the Liberal Democrats were still a significant force as they typically sit somewhere between our A and B on a lot of matters. For whatever reason, smaller parties have persisted in some specific areas despite having no chance whatsoever of winning nationwide. The Northern Isles of Scotland are committed Liberal Democrat voters, for example, even though they've not been anywhere near winning nationally for a century. The C is now a pro-Scottish-independence party that is absolutely never going to agree on much with A, and which B is going to be hesitant to work with despite a number of similar policies because B doesn't want Scotland to leave either, so A and B are looking at the really small parties to work with when they need to.

The Scottish and Welsh parliaments use a mixed system. Two thirds of the seats are appointed with FPTP, but everyone makes two votes. Your first vote is for your constituency just like in Westminster or the US HoR, but you also have a second vote for your region, a collection of about eight constituencies which also gets multiple seats. The regional seats are weighted so that parties that parties that are proportionately overrepresented get less of them, so the regions loosely counteract the imbalances of the first round. In Scotland, for example, the SNP typically wins a lot of seats in both Scottish and British elections. In the British ones this results in the SNP having a huge majority of Scotland's seats (upwards of 90% some years) while only getting a little around 50% of Scotland's votes. In Scottish parliament elections, they other parties that lost to the SNP in the first round get boosted in the regional round and it comes a lot closer to being proportional, resulting in an SNP-Greens coalition government.

Again, Northern Ireland is entirely its own thing, and this comment is already getting very long