Poor Squigs, we totally stole his thread.
To try and give some actually useful advice, give Bath a stop on your way. From London, down the M4, and you're there. Extra bonus, the M4 is really easy to get to from Heathrow airport and saves you crossing London. It's an absolutely lovely town and has some really interesting history to it and a nice spa if you're a bit travel sore. After a night in Bath, you can head north via the M5 to Birmingham as others have said. I wouldn't pass up a stop in Bath while you have the opportunity though. I wish I could plug Bristol instead, and it is a great city with plenty to offer, but if you're short of time, pick Bath over it. If you do come out west, let me know and we'll have a warm beer or high tea and discuss the correct way to pronounce the town we're in (it's Barth...)
The weather in October is the same as any other time. Changeable. Might be sunny, cloudy, wet, cold, warm, snowing, windy, hailing or all of these. There's a reason we talk about the weather so much, we have a lot of it.