Never been to any coin shops in Edinburgh...
Sadly most of my favourite LCS's in London have closed. These ones are still open and I have visited them fairly recently:
Coincraft, 45 Great Russell Street (opposite British Museum) - sells a lot of overpriced common coins (often packaged into sets), but usually have a nice selection of high-range stuff and they also often sell recent-ish
Royal Mint proof or boxed uncirculated coins and sets at below original issue price - their junk bin is worth checking out!
Gold Coin Exchange, 16 Charing Cross Road. Good selection of English crowns and gold, quality material but a little pricey.
Philip Cohen, 20 Cecil Court (near Covent Garden, close to the above). Bought a Victorian Gothic florin there recently for a reasonable price. They also had some sensibly priced
Morgan dollars and other large World silver coins.
On a Saturday the Collectors Market at Charing Cross Railway Station is worth checking out. Sells stamps, coins and currency, but not as good as it used to be.