Sorry you're having to start off the IVF journey redrollers, it is a massive learning curve. As onlyaphase says, physically I found IVF totally "doable"- what's hard is the emotional side of things.
The key thing at your age (and I'm not far behind you!!) is choice of clinic, and getting a move on. They vary widely in their success rates over 40 in particular. You are lucky living in London that 2 of the best clinics in the country for over 40s are there, namely the ARGC and The Lister.
I would strongly advise you to also look at the fertility friends website. It's UK based and there are a huge number of very friendly women on there. There are sections for over 40s, newbies, and most of the clinics and regions round the country. I learnt most of what I know about IVF there one way or another.
We had most of our treatment at our local clinic which was the preferred NHS supplier; I found it quite set in its ways. Our successful cycle was courtesy of one of the top clinics in the country which had a far more progressive approach. I guess what I'm saying is don't necessarily just go to the one on your doorstep, or the one your GP is "used" to.
You will learn that you have to be quite involved and proactive as an IVF patient. Knowledge is power and all that- particularly when you are paying. I doubt very much that you'll get anything on the NHS I'm afraid, but at least that means you have infinite choice.
If you have a sympathetic GP you may find they will help you with some initial bloodtests which can all mount up in private. Eg day 3 LH and FSH, day 21 progesterone, HIV and Hep B etc.
Hope this helps and good luck!