Got it.
I encourage you go right into London and not bother going to Heathrow, as it’s then a schlep back into London.
I can’t think of any convenient way to get from Southampton to Heathrow without an expensive private car. A car to Heathrow will probably take almost as long as a train to London. The shuttle between Southampton and LHR takes almost two hours, longer than a train to London.
If you take a train from Southampton to Heathrow, you will likely go into London (or have to change trains in Reading), then back out to Heathrow. Then you’ll have to get on a tube back into London, then out to Heathrow again, all in one day. You’ll spend a LOT of time on transport and not so much time in London, where there is always something new to see.
Plus, it’s REALLY noisy out at Heathrow.
I’d go right to a hotel in London and stay the night. There will be plenty of hotels near Victoria or Waterloo, and they will secure your bags until check in. Trains from Southampton get in at Victoria or Waterloo. If Victoria, it’s close to Buckingham Palace, etc… Waterloo will put you very close to the London Eye and Central London, but both are good locations. With a noon flight out of Heathrow, you won’t have to wake up too early.
There are quick ways to get to Heathrow from Central London. The new Elizabeth line is fast and super modern, worth a ride on its own. The Paddington Express is fast but expensive, and the Piccadilly Line goes right through Victoria Station.
Just to add, I used the Liz Line last week. The new line has eased up a lot of tube traffic, at least as far as getting to LHR is concerned. It’s really fast. I had a flight at 11:40 am. I was on the tube by 8 am and had plenty of time to spare before my flight.