As we have done this LHR/LGW tranfer quite often, I woukd just add that 3h40m is as short as I would ever allow. We have had a couple of transfers close to MCT where we have been down to the last 10 minutes due to road delays! The protected connection helps, of course, but doesn’t lower the blood pressure!
We have used both coach (National Express) and private hire. Equally happy with both, but now use YoYo Cars for the comfort and door to door convenience.