All my visits to the British Museum have been between May and September, so I have no experience with fall conditions. I was last there four years ago, I believe. I visited the museum often on that trip, arriving between 10 and 11 AM. The ground floor of the museum was nearly as crowded as the Vatican Museums until mid-afternoon or later, at which point the situation improved somewhat. The upper floor was busy but not as terrible as the ground floor.
Museum personnel told me the place was most crowded on weekends, holidays, Fridays and rainy days. You see a lot of young people in the museum, so I imagine school holidays are a tough time to visit.
If you have a timed entry ticket, you will have done everything you can to reduce waiting time. As for crowds inside the museum, going mid- to late-afternoon on Friday, when the museum is open until 8 PM, will probably make for the most pleasant visit. Just don't expect to find any food available after 3 or 3:30 PM.