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.

If money is not a concern, you could charter a helicopter from Southampton Airport to Heathrow. It takes between 20-30 minutes and you only need to get there a few minutes before departure. No security to worry about.

and train to Heathrow , with 2 changes.

It is one change- the hourly train from Waterloo to Woking (50 minute journey) usually at 20 minutes past the hour then the rail/air bus to all Heathrow terminals (except T4).
The Rail/Air bus runs every 30 minutes, 30 minute journey to Terminal 5, 45 minutes to Terminal 2, 50 minutes to Terminal 3.

For Terminal 4 and Hatton Cross you use local bus #446 from Woking Station.

I used the National Express coach for this journey and it was very easy. Much easier than the train if your destination is Heathrow.
I needed to book in advance and the coach was waiting by the ship to pick everyone up.

As well as National Express Flixbus operate from Southampton to Heathrow Terminal 5 and Central Bus Station (for Terminals 2 and 3) several times a day.