Certainly not when applying for foundation or specialty training positions, but as a current med student, I've seen that it becomes much more relevant in the following circumstances:
When applying for fellowship positions abroad (particularly for the US and Canada; certain medical schools have well-established links with universities such as a Johns Hopkins, Harvard, Yale and Emory)
When trying to set up a private practice
- I've done summer work in a number of private practices and patients frequently ask where doctors studied and I've seen them become visibly relieved when doctors have mentioned training at Oxbridge
- Almost all of the doctors I've seen in these private practices, have to emphasise studying/working in London, Oxford or Cambridge, even if only for a very short period of time
- One particular doctor I worked with was the lead for his clinic, but was the only one to emit his university from the clinic biographies because he went to a newer medical school.
I know these are obviously anecdotes, and by no means will attending a top tier university be the ultimate determinant of whether somebody is a good doctor, but I think it can be somewhat disingenuous when we say medical school prestige doesn't matter at all.
Just my 2p.