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I think I said this to @TTM earlier in the thread, but the thing that always caught me out on my first day in the mountains was lower back pain, because you can't replicate the position required to climb for 60 minutes or more in the UK. So if you can mitigate that from doing some core work, then it'll help enormously.
Also, don't lose sight of the fact that this will be a holiday, so enjoy it, and take time to look at the views.
First off, don't ever worry about average speed as an indicator in the high mountains.
I wouldn't worry too much about joining the gym. The strength you need for your upper body when cycling isn't much, it sounds like you just need to do some core exercises to eliminate the aches you get, so things like press ups, planks, even sit ups.
You'll also get some benefits from doing longer rides on the bike at the weekend. I'd build slowly but by Easter/early May, you should be able to do a 100 mile ride with relative ease, ideally being able to put together two or three long days back to back over the Easter or either of the May bank holiday weekends.
Once the evenings get lighter, I'd recommend doing a couple or extra rides a week, but at a much higher intensity, i.e. close to threshold, to get your body used to being able to push hard for 45-60 minutes.
The commute is just hours in the saddle, I really don't think you can get significant gains from commuting in London.