I just looked at my tube map. Picadilly tube from LHR to Picadilly stop; there switch to Bakerloo (?) line to Waterloo. Here you are just back of your hotel and the London Eye. This does involve some slepping of bags; whether you want to do this depends depends on what kinds and how many you have.

I have used the Hotel Shuttle (when it was included in the package) and they are good and reliable - time en transit so dependent on traffic however. Same with Air Bus (cheaper than shuttle but more expensive than tube!) but I don't know if they go to the South Bank. Might be able to find out from British Travel http://www.visitbritain.com

Enjoy your stay. I'll be there over Xmas/New Years but this time, I'm staying near Paddington so I get to take the train in - whee, a new experience!

Cheers! Jo from Sunnyvale in Northern California!