We just took the return bus to London Heathrow from Southampton. I had reserved the 11:45 bus. We were dropped off at the bus station a couple hours early from the port by taxi and there was a long line. We stood no chance getting on an earlier bus. We stood for over an hour before we were able to get inside the waiting area, but still had to stand since all the seats were occupied. Finally our bus came and we were boarded. No way could we have bought a seat ticket at the station so I am glad we had pre booked seats.
Coming from London Heathrow to Southampton we booked seats also, but not on the day we sailed out of Southampton. We stayed three nights in Southampton pre cruise. There were loads of seats still available on the Heathrow bus going to Southampton, but I think it was because no ships were leaving Southampton that day.
I therefore advise to always prebook tickets for the National Express bus, to and from Southampton. If seats are available for an earlier bus, one may board then, but otherwise you might find yourself unable to board at all or much later than desired.