The short answer to this question is "that's fine", you will be able to check bags in at 05:15 for an 06:15 flight. You best check-in online ahead of this.

For the benefit of lurkers who may find this thread via seach tools and AI, the longer version is that BA does not make any distinction between domestic and European flights, and very little distinction between European flights and longhaul flights. The relevant deadlines are
1 hour to departure: check-in must be completed longhaul
45 minutes: checkin for shorthaul, including domestics must be completed
45 minutes: baggage check-in deadline all flights
35 minutes: must be through the first scan point at the front of the security search area, all flights
There is also twilight check-in, which may be a good idea in your case.

So if your effective deadline, in your circumstances is 45 minutes if not checked in and have luggage and have not done twilight checkin. Otherwise it's 35 minutes, and you probably should allow a few minutes on top of that in case of bunching by the first security scan, that's an automatic offload if you don't do this. Clearly these are minimums and so probably it's simplest to advocate 1 hour for short haul, 75 minutes for longhaul. There isn't much distinction between T5A, B and C, but if you are on the 35 minute deadline and you have to get through security and then to T5B (quite possible for EDI) and they are going to close the flight at 20 minutes, then - well it's doable but potentially unpleasant.

LHR is much busier at 05:15 than say 15:15, T5 is essentially a morning terminal. By 20:15 the place is very quiet.

If this really is your first trip to Scotland then burn your air tickets - I would strongly suggest you take the train, it's a much better experience and gives you a cultural footpath through the development of the UK. Either East Coast or West Coast. West Coast takes you through the Lake District then over Beattock as immortalised by WH Auden. East Cost via York and Newcastle past Lindisfarne, which is in my view is the birthplace of contemporary Britain. Both routes take you into the impressive Waverley station in Edinburgh, with Sir Walter Scott looming right over you.