The Venice-Simplon Orient Express also has lovely service to London from Southampton. There are also a variety of upscale private car services that I know of, based in London that can offer private car and driver for you.
We used the heathrow /southhampton bus shuttle last summer and it was something of an ordeal. The Cunard people meet you off the plane, then herd a big group of people and all their luggage to a spot in the middle of the airport until they get enough. Then they lead you and your luggage to a bus. Then they take you to an airport hotel where you wait for a hour or two in a lounge. There was coffee and cookies if you got a good seat, but all the good seats were taken so we had to wait in a ballroom type area where there was no air conditioning. THen they load you on another bus that takes you to Southhampton (v.hot and uncomfortable and lots of traffic) Driver took a local route because of congestion on the highway and we almost lost our cookies. My husband swears that he'll never so it again.