Southampton to London via Stonehenge tour or another way?

Take a train from Southampton to Salisbury and then the Stonehenge Shuttle Bus:>https://www.thestonehengetour.info/bus-timetable/
Once back in Salisbury, be sure to walk down to see the magnificent medieval cathedral before taking a train to London (Waterloo).

Alternatively, take a train from Southampton to Winchester (which is on the direct line to London) See this historic small city before taking another train to London (Waterloo). You would need separate tickets for each leg. https://www.visitwinchester.co.uk

Don’t bother with Windsor between Southampton & London = too awkward. Do that separately from London.

Trains > https://www.nationalrail.co.uk

One of the problems is that, after a number of searches, there doesn't seem to be anywhere handy to Salisbury station to leave your luggage while at Stonehenge.

Stonehenge and Windsor Castle are nowhere near one another, so best to not try to do them both on the way to London.
Salisbury and Stonehenge are not far at all from Southampton.
Salisbury is a lovely medieval town.
If you could spend one night in Salisbury after leaving your ship, this would allow time to see Stonehenge and Salisbury Cathedral. Then you could be on your way to London and do Windsor as a day trip from London later during your stay there.

You said you can't find tours that have availability going to these two destinations.
Have you looked at your desired dates at the website for Evan Evans tours?
These tours leave from London (Victoria Station area), so you would just go on to London, check into your hotel; this way you are rid of your bags.
Do this as a day tour sometime during your stay.
https://evanevanstours.com/tours/day-tours-from-london/windsor-castle-stonehenge-and-bath-tour/
https://evanevanstours.com/

If you are seeking a private car service to take you from Southampton to Stonehenge, then onward to Windsor, then to London, price being no object, please say so.

I did as James suggested. Walked my luggage off Cunard, taxi to train station, train to Salisbury. It was early, so dropped luggage at pub across the street, walked into town for breakfast and came back to train station to pick up the HOHO bus to Stonehenge (that was the bus available December 2018). Returned to Salisbury for a walk around town and the cathedral. Took National Express bus to Heathrow.

National Express only run one bus a day now from Salisbury to London at 0950, and none at all to Heathrow.
Timetables were much rationalised when services restarted after the Covid closure.
Heathrow services now run from Southampton and a park & ride site at Winchester (not the city centre).