Gatwick trains run around the clock to victoria, and a bus/cab would do the trick to knightsbridge. (oh yes, if someone did miss the point, i land @ Gatwick, and may head to Heathrow in the morning)

Checkin in at 2am would be less stressful to me since i dont have to bum around in the airport for 4 hours, can offload my luggage (but i guess i could store without checking in anyway) and also need not listening to mumblings during checkin in jetlag mode.