We have sailed out of Southampton many times. After a Princess transfer had to take a 45 minute stop at a highway rest stop because the driver had gone 4 hours without a break (it is law in the UK) we looked into private car services.
We have used Smiths for Airports numerous times for both going from LHR to Southampton and from Southampton to LHR. The cost is reasonable (similar to 2 Princess transfers). Driver meets you at the airport or cruise terminal, helps you with your luggage, and drives you directly to your destination. Vehicles are clean and owned by the driver. Most drivers are ex-military. Most times transfer is 60-75 minutes (and no waiting for the Princess transfer bus to fill). You can check them out on trip advisor.