4.20 flight, arrival at Heathrow by 1.20.
So work backwards, and it's not clear why you are changing taxis at Gatwick.
Lewes to Gatwick 33 miles- one hour should give you time to spare;
Gatwick to Heathrow 39 miles- another hour albeit round the M25, add on half an hour for bad luck.
That's 10.30 from Lewes.
If you meant to take the coach from Gatwick to Heathrow use either the Oxford Bus Company the Airline,
or National Express.
1 hour 15 minute journey- the Airline leaves at 1105 and 1205, arriving at 1220 and 1320;
National Express at 1110, 1120, 1205 and 1210. They are scheduled at an hour.
So even then between 10 and 10.30 from Lewes gives you a nice journey, which shouldn't be stressed.