Americans commonly use "tyre" and "wheel" interchangeably. Drives me nuts especially in motorsports where the distinction is important. But it is what it is. This was a case of a wheel coming off.
Not sure if I missed your joke. But this is probably a joke based on the spelling of the word being British. So Americans use tires and the British use tyres.
I wasn't joking, I'm not going to use the American spelling just because I'm talking about how Americans use the word. They really do use "wheel" and "tyre" interchangeably.
I'm an American, and I disagree, mostly. I've never heard someone say a "flat wheel"; it's always flat tire. Also if an American says a wheel came off when they mean a tire, they misspoke/misunderstand what they are talking about. There are plenty of Americans that know next to nothing about their vehicle despite it usually being the most expensive or second most expensive thing they own.
Disclaimer: Midwest USA here, can't speak for other parts of the US.
I can't think of anyone other than the most archetypal ditz (think, 'Friends' character, Phoebe) who would conflate the idea of the rubber part that touches the ground with the metal or other hard material the first term is mounted on. (Naming abstractions removed for clarity.)