"London" in theory has several airports, Heathrow, Gatwick, Luton & Stansted, plus a few smaller one's like London City. But for your purposes the two main one s are Heathrow & Gatwick, both quite a way from central London but you can catch a train bus or taxi into the city. I would say (not that I have done it) that the train may be best/cheapest, they run from inside the airport I think. As far as being closer to Southampton, they are both around 100 miles away (average) so that is irrelevant. if you are coming all that way I would say have a look at London & if time the Southampton area has things to see, like Winchester, Portsmouth, New forest, Stonehenge, Marwell zoo-park is interesting & there are good West-end /Broadway standard shows at the Mayflower theatre.