I take it that by saying "Buckingham" you mean the Queen's residence? The name for that is Buckingham Palace.
British names are not generally shortened or given just one word of the whole name. There are many roads in and around London with Buckingham in the name, an entire county between Milton Keynes and Oxford, and the name of the county town.
Unless you are going in the summer months when the Queen is away (her 90th birthday now) you can only see Buckingham Palace from the outside at a distance. That won't take long. Even if you wait the 2 or 3 hours after standing outside to see the Changing of the Guard there that's only 2 or 3 hours.
Guildhall Gallery is an hour or so at most.
So that leaves 2 and a half days for the Natural History Museum.
If you want to spend lots of time there I would suggest the Crown Plaza or Holiday Inn in South Kensington or Earls Court.
If you really want to spend most of your short time more central you could still stay there and just use the Tube or buses. London is pretty spread out so that you can't see everything in a short walk from anywhere.
If you want to be closer to the Palace you might try the Holiday Inn at Victoria or the Mayfair Holiday Inn across from Green Park and Piccadilly, just down from the Ritz.
Expect to pay big for the closer in ones, either in Reward Night points or money.