London: Baker Street for the Sherlock Holmes house and museum is small but worth a visit.
The V and A, Science or Natural History museums in South Kensington
The Cabinet War Rooms (where Churchill hid in his bunker and ran the war from) near Green Park and Buckingham Palace.
Lambeth for the Imperial War Museum
The Mummies and Egyptian stuff in The London Museum at Kings Cross/Euston
Kings Cross also has the Harry Potter platform and 3 quarters which now has a Diagon alley. Good for kids.
Covent Garden and Carnaby Steet for shopping and mooching.
Don't go to Madam Tussads as it's a rip off.
Central London is not very good for food, hence the bad rep we get worldwide. For good food and international cuisine Edgeware Road near Paddington, this is very multicultural and not populated by many visitors to London, you can get some real tasty kebabs here. I also had a really good haircut from a Turkish barber in Edgeware Road once that cost all of 6 pounds and took him 5 minutes. Brick Lane and the Spittlefields market near Liverpool Street, if you like curry is popular and tailored towards tourists. Although be careful in Brick lane some places are better than others and some places are not good at all.
Some rare treats: Check out the Old Operating Theatre down on the South bank
http://www.thegarret.org.uk/ The Golden Hinde is along the river as well as The Clink museum (London's first prison).
Also you have The Tate Modern down on the South Bank if you like contemporary art, there are usually other exhibitions along there and you can see Big Ben and Parliament over the river. A nice stroll with fish and chips can be had from a pub near the Golden Hinde.
With limited time I would go and see the Sherlock Homes house and the Cabinet War rooms as they are fairly central and easy to get to on the tube. If you like science the science museum has the Rocket in there and the first steam engine by Watt.
If you like rare books and magic and druidic stuff (yes magic) look no further than Treadwells at Tottenham Court Road
http://www.treadwells-london.com/london-shop/
or close to there and lighter Forbidden Planet for comics:
http://forbiddenplanet.com/stores/
There is a GW on Oxford Street as well as Covent Garden (tiny one, lower level to the touristy market area) and other places if you wanted to save on the Nottingham trip.
Hope this helps. I used to work for London tourism and these were some of my favourites.
Outside of London yes, Cotwolds/Bath and Devon. Devon you would need a car for and more time but the Cotswolds and Bath you can do as a day trip from London (bus tours, 2 days might be better).
Stratford Upon Avon is possible for a day trip if you like Shakespeare.
Scotland I'm still meaning to visit.
