You may not get the best possible price if you walk in off the street onto retail premises looking for silver or gold coins, but if you're in London with some money burning a hole in your pocket and get the urge to pick up a sovereign or a silver britannia....
THE CITY OF LONDON
John Haynes is a tiny jeweler in a warren of alleys near Bank station. They'll sell you sovereigns and krugerrands, but I don't think you'll have much luck finding any silver.
THE WEST END
If you're in the Covent Garden/ Strand area: ATS Bullion right next to the entrance to the Savoy Hotel:
Harrod's deals in gold bullion. Just go down to the basement.
I've never bought from them, so I have no idea how easy it is, or what their prices are like, I just know they sell it.
... is on charring cross road near trafalgar square.
Was looking for silver but found no luck.