If you want to see London and Britain, you're going to need a very long time: London, as the top Alpha ++ "world city" on the Global Power Index (New York is in second place) is vast - it's also made up of two cities (the City of London and the City of Westminster) plus numerous towns, each with their own character. The greater metro area is home to more than 14 million people, and it's been a city for 2000 years... there's a lot to see, and experiencing even a small percentage of it would take a long time.

Then there are the other cities: Cambridge and Oxford (Oxford, though as a Cantabrigian it pains me to say it, is much prettier), Manchester, Salisbury, York, Edinburgh, Lincoln, Canterbury, Norwich, Exeter, Bristol and more, as well as hundreds of historic towns. The good thing in this respect is they're all much smaller than London, and you could see most of any one of them in less than a week. If you want to see the historic sites outside of the cities, there are thousands of megalithic sites - Stonehenge is most famous, the Avebury complex is more impressive (a stone circle much bigger than Stonehenge, plus the West Kennet Long Barrow, Silbury Hill and more)... fortunately, they're only a few miles from one another and there's plenty more nearby (Weyland's Smithy barrow, the Rollright Stones and - IMO, one of the most spectacular sites in Britain - the Uffington White Horse. Every part of the country has its own "feel" and traditions: Cornwall, for example, is completely different to East Anglia, and Kent is completely different to Yorkshire. Wales, Scotland and the Isle of Man are completely different to one another and England, and even have their own languages (in Wales, especially North Wales, you're likely to hear Welsh spoken. It's very different and much older than English - or, as the Welsh would say, "mae'n wahanol iawn ac yn llawer hŷn na'r Saesneg" ).

The good thing is that the entire of the UK isn't very big - you could drive from Land's End in Cornwall to the far north of Scotland in less time than it'd take to drive across Texas, though zig-zagging around to see everything along the way would increase the distance many times over.

If you're going to be based in London, spend a couple of days checking out the museums, galleries and a palace or two and St. Paul's, maybe take in a West End show. If your tastes are a bit more sub-culturey head up to Camden Markets in the tube.Get out to Dover for a stroll along the white cliffs. Head out to Stonehenge. Either Oxford or Cambridge are lovely.
If you have more time to travel further afield, the coast around Cornwall is beautiful, and Edinburgh is great for a few days and you can head from there up to the Scottish Highlands - the landscape around Fort William, Ben Nevis and the Isle of Skye is pretty special.
Just don't go looking for wilderness. Britain doesn't have any.

I love castle hopping , especially in Wales . I had a friend who owned a car and was on a PCS in London . We drove from London to Wales , then in one weekend located 13 castles , saw 12 , and got to spend two nights in one . In London , I'd recommend getting a transportation ticket/pass . It covers the subways , railroad , and buses . Think of London with a target placed over it . You can buy a ticket good for , let's say a week , for which ever of the circles you chose . You can even chose to stay mostly within the inner circles (costs less) , then if , for instance you decide to go to , to the gardens , you can pay an additional fee to extend your ticket to an outer circle for one day . I'd also recommend getting the rail to and from the airport to London . Much less expensive than a taxi . I'd suggest looking for a map that has what you want to see marked on it . I found one that showed where castles were located before I went . Great investment ! I was also a member of Mensa , so chose to go to a Mensa event while I was there . The guest speaker was Terry Prachette , a favorite author , and the event was being held at a castle , so we only had to pay a group rate for our stay at the castle .

London has some great museums and art galleries, it’s well worth spending a few days on those. Pretty much all free of charge too.

London is great, but do not spend all your time there. There is a lot more to the UK, and if you have never been to Europe before and you really want to widen your experiences, then of course you need to get out of the cities with their international culture to find genuine and different cultures. Scotland and Ireland are great, but do not forget Wales, Cornwall/Devon, East Anglia (very close to London.) or the North of England, all of which are not as far from London and very different.