You can always take a black cab, but as all airports except London City Airport are quite far, you would be looking at 1 - 2+ hours drive and £100-£200 + fare. You can book a private shuttle service, which is cheaper. Best to use for that is London Airport Shuttle Service | Book Airport Taxi Online. You can also use Uber.
However, if you would like to get around like a local, here’s what we usually do. First of all, we use CityMapper for all our commuting needs - it shows the best possible routes, any delays, costs etc. All of the below is based on that all the trains are running and there’s no suspensions. weekend railway works or people on the tracks. Always check CityMapper before committing to a route, as all of the aforementioned may result delays also in to your journey.
London has 5 airports, each having a connection to a different train station in London.
Heathrow (LHR):
Option 1 - walk to the knightsbridge tube station (5 mins) and take the Piccadilly line (dark blue) to Heathrow - Approx. 50 mins
option 2 - get a cab / uber
Gatwick (LGW):
Option 1 - Walk to South Kensington station (10 mins) and then take the district line (dark green) to Victoria station, then from there take the Gatwick express (approx 50 mins)
Option 2 - Walk to Knightsbridge station (5 mins) and take the piccadilly line (dark blue) to Kings cross - from kings take the gatwick express - approx. 1.5 hours
Option 3 - cab / uber
Stansted (STN):
Walk to Knightsbridge station (5 mins) and take the piccadilly line (dark blue) to green park. From there take the Victoria line to Tottenham hale where you can change to the Stanstead express - approx. 1 hour 15 mins
Luton (LTN):
Walk to Knightsbridge station (5 mins) and take the piccadilly line (dark blue) to Kings cross / st pancras international - from st pancras take the EMR to Luton airport parkway. From there you can either walk or bus to the airport
London City Airport (LCY):
Walk to Knightsbridge station (5 mins) and take the piccadilly line (dark blue) to green park. From there take the Jubilee line to Canning town, and change to the DLR which goes direct to the airport
Check CityMapper for real-time routes as trains may be cancelled, delayed, etc.
