Chiltern services to Aylesbury depart from Wembley Stadium station. I've no experience of how quick it is to get on the platforms, but from there it's one stop to Northolt Park followed by a 25 minute drive to T5. You can travel a couple more stops as far as Denham to reduce the driving distance, although Northolt has more route options in the event of issues on the M25.

That's probably the quickest option by quite a way, but the hardest to find bag storage.