The actual journey time on the National Express is somewhere under an hour. So long as you get through the UK Border quickly at LGW, and baggage reclaim is swift (both can be a bit sticky) then 4 hours is fine. It can just about be done in 90 minutes with everything in place. National Express would charge £60 for all of you, thanks to the kids' ages, so the pricing is in line with Uber or whatever. No problem with baggage space and the kids may like the views from high up in the coach. You can buy from the machines (by the exit area) once you get to LGW, based on the time of getting there. If you have over 2.5 hours left then the coach is fine, if under 2hrs then think about ringing one of the numbers in the other threads (and don't have about!).

However! To be honest, in that situation I would suggest flying with Easyjet from Gatwick to Paris, it would be a lot easier. You cannot through check luggage but the Gatwick Connect service in the LGW luggage hall will through check it in the BA to U2 direction (but not the other, unfortunately). Also, if BA continue to expand LGW's services to compete with U2, it would not surprise me if BA add a Paris service from LGW by the time you get to travel.