September is often nice and warm, in the 80s. But this is England, the weather can turn on you just like that
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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.