I personally have never heard of a student transferring between universities during their degree, but I suppose it may be possible.
I think the reason for this is because each degree is specific to the department and university, so someone with a Psychology degree from say, Leeds, may have studied different courses to someone with a degree from Bristol.
As far as I am aware, the only way to go to a different university from the one you started at, is to quit the first course and start all over again somewhere else, thereby wasting a couple of years of your life!
Another option, like bookgrl says, is to try for an International exchange program. I'm from the UK, but did a one-year exchange program with the University of New Mexico during my third year. However, only certain degrees courses allowed you to study abroad, but it seems to be easier in the US - a friend of mine at UNM spent one semester studying in the UK and then another studying in Brazil!
Something else to consider is that classes in the UK are likely to be at a different level than the US. Although I had just finished my second year of undergraduate study in the UK, I was put into senior and PhD level classes at UNM, whereas exchange students from the US (at Junior level) may take first and second year courses in the UK - basically, the school systems are different and we usually end take courses equivalent to college freshman and sophomore level (at high schools or further education colleges) before we even start university in the UK.
