if you are coming to London for a few days precruise then Gatwick is the better choice, the gatwick express train takes you into London Victoria it's a fast line my wife often used it when there were problems on the normal commuter line into Kent. Then I would pick her up on my way home at the airport. Heathrow is closer to Southampton but not a lot and taking traffic into consideration no quicker to Southampton at peak times.
The main thing to think about is how to get from the airport to the ship if you are going direct to the ship have a look at National Express coach company for costs.
You can also get prices for hire cars with driver they are like cabs but are not allowed to just pick up at the road side. You can contact them by phone or email they will give you a price to take your party and luggage it is probably cheaper than the train and a lot easier door to door