We're just back from London a couple of weekends ago. The Shard is amazing, and well worth a visit, and the British Museum has a greap exhibition of artefacts from Pompeii and Herculaneum.
Orcs Nest is on Earlham Street just off Shaftesbury Avenue, and is a rather decent games shop. It's very close to the London Graphics Centre on Shelton Street, which is a fantastic art supplies shop. It's also just along the road from Forbidden Planet, the big comics and merchandise shop on Shaftesbury Avenue. All are a five-minute walk from Covent Garden.
Leisure Games is THE RPG shop in London, but it's a bit of a journey out to Finchley on the Underground. The shop is, IIRC, pretty close to the station.
Been years sice I lived in Liverpool, so I can't help with the games shops, but check out Williamson's tunnels - absolutely brilliant. You can also get a tour round in a bright yellow DUKW (on land and in the water in Albert Dock), but be warned - two of them have sunk in the past few weeks!
I'm on the wrong side of the country, so sorry I can't help with anywhere to sleep.