If the flight's on time you should make it with no sweat at all. LGW has a trannsfers system so you don't need to clear customs / immigration. You WILL need to change terminals (S to N) but this takes about 5 minutes.

I should expect that your bags will be fine as well. LGW is much better than LHR for US - European baggage transfers. At LHR this normally involves a T4 - T1 transfer, and many bags don't make it in the itme available.