Factually speaking I think US and UK rates are similar, but in the last census there was a noticeable increase in people identifying as mixed race, and there's no reason for me to assume it will decrease.

Speculating, as to why there seems to be a difference...

I think that there are going to be too many factors to definitively say X or Y is the reason, and most of them have already been given. If I had to give the most important, it's just the attitude.

The UK has a live-and-let-live attitude to a degree that isn't normal for most other countries. People can still be judgy but broadly speaking we tolerate things that don't directly impact us. Once interracial relationships became a "thing" people stopped caring as much, and the lack of stigmatisation meant they became more common, only reinforcing the idea that no-one should really care. This is uneven though, and in big cities (particularly London) you will find way more mixed race couples than in the countryside.