We did this with an overnight. If it would have been anything else, we would not have made it. First off, we hade a one hour delay in AMS. We then had a 1 1/2 hour wait for our luggage in LGW. We booked a private driver who charged us extra for the long wait. We then got stuck in traffic and he went off the highway to get to LHR hotel. He got lost.
All in all, we should have arrived at LGW at 11 am and figured we would get to the LHR hotel about 1- 1:30 pm. We arrived at 5:30 pm.
I would not even think of trying to make a 3 hour connection. Too many things can go wrong.