Here is what we did. It was painless and cheep.
We arrived at London Heathrow @ 10:30 local time.
We took the bus to Southampton. There is a bus at 11:50. No changes; 1 hr 50 min. Arrives at 13:50. Catch it at the Central Bus Station at Heathrow.
£48.24 total for 4 ($89 USD + 10% Tip for the driver)
Drops off at SOUTHAMPTON, coach station, Harbour Parade, Hampshire — GB – Southampton
We took a taxi from there to the Novotel Southampton. It was 5 GBP + tip 10% ($10 USD).
The next day, we took a taxi to the pier where the ship is. Left around 1100 or so. Taxi was 10 GBP + 10% tip ($20 USD).