One of the things I've enjoyed while I've been in London was climbing up the inside of St Paul's dome to the Golden Gallery right at the top - it's around £20 per person but was worth it.
https://www.stpauls.co.uk/history-collections/history/explore-the-cathedral/climb-the-dome
I also enjoyed climbing to the top of the monument to the Great Fire. It's a bit like climbing up the inside of a giant candle, but again great fun and much cheaper.
http://www.themonument.info:
HMS Belfast is worth a visit if that's your sort of thing - I enjoyed looking around it:
https://www.iwm.org.uk/visits/hms-belfast
I'd also suggest the Tower of London and the river walk around around there. The Tower is right by Tower Bridge, which is always fun to cross. Walking back the other side and crossing over the wobbly bridge makes a nice walk too - you'll pass lots of old buildings, a reconstruction of Shakespeare's Globe Theatre and end up by Tate Modern, which is housed in the old Bankside power station. Cross the wobbly bridge and you're back at St Paul's.
https://www.hrp.org.uk/tower-of-london/
http://www.tate.org.uk/visit/tate-modern
https://www.visitlondon.com/things-to-do/place/442404-millennium-bridge
The Royal Observatory out at Greenwich is fun too, and Greenwich is really nice. I know it's an artificial line unlike the equator but I always find crossing the Greenwich Meridian fun.
https://www.rmg.co.uk/royal-observatory
Even better you can get to Greenwich on a water taxi:
http://content.tfl.gov.uk/river-bus-tours-map.pdf
Cutty Sark is there too, which is fun if you like old boats:
https://www.rmg.co.uk/cutty-sark
Central London is really quite small so you're never far away from stuff. The Tube is pretty good too.
Guitar shop wise most of the central London shops are on Denmark Street or very close on Charing Cross Road. The Tube stop is Tottenham Court Road. Not a lot of PRS though. I also like New Kings Road guitars, which is vintage stuff.
Never been to Andertons, but it's in Guildford, which is around 30 mins out of Waterloo on a fast train (and an hour on a slow one).
I think the best PRS shop in England is probably World Guitars, in Stonehouse. It's a long train ride out of London (but the shop is near the station) but could be a stop off if you went to Bath, which is lovely.
If i can think of anything else I'll add another post. But it's a bit like suggesting films to watch - your taste might be different to mine.
Enjoy your trip
