Just got back from the Savoy. I ended up not taking the tube. One of our bags was 70lbs, and I just didn't want to bother lugging that around on transit. Funny, we weren't nearly that heavy going over--must have been too much shopping in Paris and London.
I ended up using justairports.com for the transfer. Total cost was 36 GBP for an estate car. Nice new vehicle, showed up on time, called just before arrival, and got us there on time.