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In London, there are lots.
In Soho, there is Reckless Records and Sister Ray, right across the street from each other on Berwick St.
There are a few others in the neighborhood, including Sounds of the Universe (where the Stones launched the Blue and Lonesome album with the 100 numbered lithos). I think the blue tongue is still in the upstairs window.
(Plus nearby is FOPP at Cambridge Circus, which is owned by HMV but there are no used records).
In Camden, there are several record stores, including Wild Horses in the market itself. The guy who owns it is a huge Stones fan. There is another one about 150 meters away, next to Dingwalls, but I can't remember the name. There is also Sounds that Swing on Parkway and Off the Floor on Inverness St. All within a couple minutes walk from each other. There are several more places that sell vinyl in the area, some of it specialty, like Reggae or Punk. Old original Stones records are expensive. Some bootlegs are reasonable.
In east London there are quite a few, like Rough Trade and another Sister Ray. Plus on the weekends there is a place off Brick Lane with thousands of used records for sale.
You can also find used vinyl at many of the street markets around London.
Can't help with Edinburgh, though I did buy a Stones record up there about 4 years ago.