Does anyone have any advice about the best way to travel from the port in Southampton to Heathrow?
Define best. Cheapest? Fastest? Fewest connections?
Fastest, and most expensive, with fewest connections would be a private car/driver from the cruise terminal to your Heathrow terminal. Or the ship transfer by coach. Then there is the option of local bus or taxi to the coach station and the direct coach to Heathrow coach station. Or bus/taxi to the train station and train to Heathrow , with 2 changes.
Take a taxi from the port to the bus station. Tons of taxis will be at port. Take National Express Coach to Heathrow. I’ve done this twice and it works great. Minimal luggage handling compared to taking train as driver loads it under the bus. Much cheaper than cruise line transfers.
This is a popular method so buy your tickets in advance. This route can sell out on some trips on turnaround days. Any trip after 9 am should be fine if self assist luggage off ship. If you use luggage handling 9:30 am.