Good question. Lets hope not before Saturday, as England has closed airspace since Icelands unpronounceable Volcano took up smoking
According to BBC World News, the Eurostar folks have laid on a number of extra trains, but it would be a good idea to reserve immediately.
"Best" is a very subjective term. A private helicopter from the lawn of your hotel to your private luxury jet at Nice airport would be one idea of best. OTOH if you could arrange to have the roads cleared a Bugatti Veyron might be interesting in place of the helicopter...
But seriously easyJet from Nice airport to one of several London airports would probably be the least expensive and among the quickest of your options. You could check AF and BA as well but they would likely be significantly more expensive. Be sure you understand easyJet's baggage policies and limits all their added charges before you book. Excess baggage fees can quickly make that cheap fare look pretty sick. Even BA and AF have more restrictive baggage policies in Economy than many US travellers are accustomed to.
By train it would take you most of 11 hours.
The sooner you book the less it will cost you to travel whether by air or rail. Fares, especially on budget airlines, climb pretty steeply as the travel date gets closer.
Sorry to add another factor, but in going to London consider the cost and convenience of the transport you will use from the destination Airport to your ultimate destination there. 'London Luton' ,as Easyjet style it, is almost closer to Birmingham than London with correspondingly high train fares which may be higher than the cost of your budget air fare. Heathrow has probably the best choice of onward travel connections. London City Airport is by far the closest to the city centre.