Take the tube to central London. Change at Earl's Court to the District and Circle Line. Bayswater is near Paddington, just a few stops from Earl's Court.
The easiest way to get to Paddington from Heathrow is using the train - it's an express one & take 15 mins. I think it costs about 15 pounds.
You can get there by tube but it'll take you a lot...
Take the Gatwick Express (overground) to London Victoria Rail Station. Transfer to Victoria underground station (same building) and take the Victoria Line north to Oxford Circus tube station....
There isn't a direct one. As you are travelling between airports, I assume you've got baggage and will want to avoid long esculators and stairs. I recommend you take the London Underground Piccadilly...
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