To be honest I don’t think that many British people hold Americans in contempt, I just think that those who do make a lot more noise about it. It seems like theres a bit of a fashion in the UK and Europe, for US bashing, and generally I find that it’s people who have never even been to the US that do it the most.
I guess it’s natural that people will have a bit of amnimosity toward the only worlds super power, and im sure that some academic somewhere will tell you that its to do with the perception of an individuals Identity being threatened.
At the end of the day, as a Brit I feel that we have stronger connections with the US rather then a lot of the countries in Europe, and in the admitedly unlikely event that we ever needed international help, im sure the US would be there for us a lot sooner then many European countries!