We always fly into London from Australia, direct, arriving around 5am
or 6am, after a 22 hour flight (all being well with touchdowns and takeoffs
in Singapore or Bangkok!).
We usually shower and change at the airport, have breakfast and read a
local newspaper, then pick up our hire car and set off for the day, often
driving up to the Midlands for business or to friends in Oxford or to
relatives in London.
It works for us, putting us onto the same time as the locals, and we never
have any trouble with jet lag that way.
Of course, our flights come into Heathrow but that shouldn't make any
difference. The basic facilities must be available at Gatwick too.
I'm not saying that it would work for everyone - just saying that it can
be done, and has been done many times, after a quite long flight.