Hiya,
The competition varies between schools, London has more places but then a lot of people apply there! I looked a while ago and the applicants per place ratio varies between 7 and 20 per place. I think it was highest for the scottish Unis but it varies year on year (people think 'oh god everyone applies to ....' and applies somewhere else so the competition shifts!)
This year I know Nottingham had 9 applicants per place and stupidly I applied thinking cos it was new there wouldn't be much competition!
I only know the statistics for the college I applied to at Cambridge (20 interview candidates for 2 places, don't think many people were rejected without interview) but the competition varies between colleges, and of course there's 'open' applicants.
Applications are declining slightly and the vet schools are offering more places than they used to but entrance requirements get higher and higher as the standard of applicant (amount of work experience, extra curricular activities, A/A* grades) rises.
Overall I think the amount of places is about half the number of applicants, but a lot of people are put of from applying by the high competition and entry requirements.
I reckon Law at the top Law schools is probably more competitive but then there are more law schools so the ones lower down the tables are probably not as competitive as vet school. However I think you guys have to compete more fiercely for jobs at the end of it!