September is often nice and warm, in the 80s. But this is England, the weather can turn on you just like that
The train from London to B'Ham (pronounced Brum) should be about 90 minutes to two hours from London Euston Station IIRC, I haven't done that trip for many years (google will help here). As for intl travel and CV19 procedures I cannot help you there, not been out of the country for two years, they usually advise arrive 2 to 3 hours prior to departure but that was 'before'.
So much to do in London. Where to start? Museums (Imperial War Museum is good, many are free or a small donation), theatre, shopping, eating out, The Tower, Buckingham Palace. Some of the guided tours are surprisingly good even for a local, giving a lot of little bits of strange but quirky info on the city. It is one of the best cities in the World and has a very varied and sometimes shocking but always interesting history. Hidden London (hiddenlondon.com//) and (Secret London secretldn.com/) are good resources if you like the different sort of tour.
For B'Ham you'll need to ask a local. The only thing I know there is the Ming Moon restaurant, nice Asian/Oriental food buffet.
If you have family here they might be able to suggest more suitable options as they'll know your tastes and interests better. If they're a bit more extended than that, some idea on your likes etc might help.