UKC

Looking for inspiration with something to do! 

Days (in UK at least) a bit short to cycle tour / hike, not quite cold/snowy enough to go skiing.

Do you have to stay in the UK?

Shorter road bike days/ climbing wall sessions and then learn a new skill. Crochet, woodwork, sewing, stone masonry, whatever takes your fancy! I always think November is a time for doing crafts!

No, though can't be away for all of it (have 8-year old daughter who doesn't live with me).

If you've got to stay in the UK, I'd get out bouldering or on the hills whenever it's nice, but that won't be often. I'd go to the wall alot, and take up some additional training like yoga. And I'd do some stuff for my mind, specifically learning meditation and getting into a regular habit, plus learning something you're really interested in. The best thing for that I've found is https://www.thegreatcoursesplus.com which I highly recommend.

But I'd much rather not stay in the UK at such a horrible time of year.

True; though seems a wasted opportunity to do something I couldn't otherwise. 

How about coastal walking, might catch some good storms, easy escape to warm hostelry in case of bad weather.

Checked your location, from London anywhere from kilve in Somerset to the east end of the Jurassic coast, or Pembrokeshire & Ceredigion will be suitably wild and scenic.

Climbing in Northumberland in November can be interesting.  I remember having so little sensation in my figures I had to look at them to make sure I was holding the rock.  It was nice and dry though - the swift breeze dried by icing the moors up saw to that.

Go somewhere warm for a few weeks and go diving 

Is it too short to cycle tour?  You should have 8 hours of daylight which is enough to get some distance in especially if you are willing to do some in the half light...true it will probably be sh*te weather, but doesn't that make it more rewarding in a way, as long as you can find somewhere to dry your kit at the end?  Bivvying out though but tired bones and a good book in some comfy hostelry sounds like a nice way to spend an evening?