The UK CAA have an archaic system where an initial Class 1 applicant had to make the trek to the Aeromedical Centre (AMC) in Gatwick to go from room to room to undergo a battery of tests. Great fun if you lived in Scotland.
The FAA makes no distinction between initial and renewals. You either meet the standard or you don't. In the US, there are "regular" AMEs who can only do class 2 & 3 medicals and Senior AMEs who can also do class 1. International AMEs are automatically designated as Senior AMEs so they can all do all classes.
Re: student pilot/medical certificates - up until 2016, AMEs could issue a combined student pilot/medical certificate. That is no longer the case as student pilot certs have now been taken over by Flight Standards Service, which makes more sense.