London Visit - Getting from British Museum area to Wimbledon and on to Hampton Court Palace and Back

It's a shame you don't want to use the bus as the quickest way is to get a bus to Waterloo (59, 68 or 168), or walk 5 minutes to the Kingsway stop at Holborn outside The Shakespeare's Head) and there are several others available too. You could walk to Waterloo as it's only ten minutes more than walking to Temple or Embankment for the District Line. The trains are quicker than the Underground and some run fast to Wimbledon.

Another possible route to Wimbledon is a quick walk, tube (Piccadilly line one stop to Kings Cross/St Pancras) or (yes) bus (number 59) to St Pancras International station and relatively quick Thameslink train to Wimbledon. Might end up a tad quicker.

If you end up taking the district Line to Wimbledon then Southfields is actually the closest stop, just a 5 min walk. After your tour perhaps a quick taxi ride to the mainline station and then a train to Hampton court. Check train times, it's not a hugely frequent line, you might be waiting 30 mins otherwise. On your way back take the mainline straight through to Waterloo. Personally I love walking over Waterloo bridge, best view of London, but if you don't fancy a 40 minute walk back or a bus you should just take the northern line tube to Tottencourt Road

Here is the trip planning site at Transport for London, the government transportation agency: https://tfl.gov.uk/
If you practice using it now, it can be helpful when you are in London. Google Maps is getting better at planning routes in cities too.

This is a map of all London's Tube and rail lines (from the above-mentioned TFL website), if you like to navigate by maps: http://content.tfl.gov.uk/london-rail-and-tube-services-map.pdf
Wimbledon is at the top of square C5, and as you can see you can get there either on the District Line (tube, green), or train from Waterloo (red dashes), or other lines from other suburbs.
Hampton Court is in square A5, at the end of one of the branches from Waterloo.
An Oyster card is valid on all these routes.

You say "near the British Museum", you need to find out which is your nearest tube station. Depending on exactly where, this could be Russel Square or Holborn. Hotel websites should give this information.

P.S. why are you avoiding buses? They are slower than tube/trains, but no other disadvantage. I would definitely prefer a bus to an expensive taxi.

Thank you all for these great ideas - I really appreciate the thoughtfulness and help.