Wise move to come a day early! Yes, if you fly in the day of sailing, and you booked door to door, SS will provide the transfer to the ship. If I were in your situation, though, and coming in one day early, I would arrange for a private transfer or take a cab directly to Southampton. You're going to be very tired when you arrive that you won't be able to enjoy London and you are going to have to pay for a transfer to Southampton anyway the next day. You will also avoid all of the London traffic in and out of London.