Any suggestions on how best to arrange transfer from London to Southhampton to catch a cruise ship? If possible, we'd like a transfer that stops at one or two places of interest along the way instead of just getting on the bus in London and going straight to the port. Thanks!
Even by private car service I wouldn't be wanting to stop off on the way to port if sailing the same day.
My focus would be on reaching the port. I'd rather be on the ship as soon as embarkation starts soon after noon, than suddenly getting stressed due to a road closure caused by an RTA or other reason, and suddenly end up tight for time. You can sightsee on the way back to London after the cruise.
If you want to sightsee on the way down then stay in Southampton the night before the cruise.
Cruise Critic has a Ports of Call forum for Europe. The first sticky on the British Isles subforum is a lengthy thread dealing specifically with transfers between London and Southampton (both directions).
https://boards.cruisecritic.com/forum/148-british-isleswestern-europe/
I agree that it's not a good idea to be trying to sightsee on the way to the ship on embarkation day.
The road journey out to Southampton can be interminable and can open up the possibility of delays for accidents etc.
The train from Waterloo to Southampton takes around 75 minutes, from where it's a simple cab ride to your ship - much the best way in my view.
Forget stopping along the way. Take the train from Waterloo to Southampton. Then a cab to where your ship has docked.