Since I saw this post, quick reply:
1. Passport control and baggage in LGW could easily take 30 mins.
2. The fastest way to reach LHR from LGW would be train- Gatwick express. LGW-London Victoria (30 mins), from there an underground train or cab to London Paddington (15 mins minimum) and then Heathrow Express to LHR (15 mins).
Total travel time is 1 hour without waiting time, let us say waiting time is 1 hour total including passport control and bags. Since you have almost 4 hours between flights you should be OK unless there is a delay on the LGW flight.