I'm over 40 and starting privately at King's Fertility. Entitled to one NHS round but coughing up for the sake of being able to do it straight away and on my preferred timeline.
I chose them because they're decently rated and cheaper than others in London for what I want to do. You can do appointments at either Denmark Hill or Canary Wharf. I'm doing a 3 cycle embryo batching package with short protocol, high stimulation and PGTA testing. The costs are: £10,395 for the underlying 3 rounds (£13,500 at Lister), £1,355 per round for the drugs and £1800 per 4 embryos sent for testing.
Where this is cheaper compared to other places: 1) all blood tests as needed throughout the cycle and advanced embryology are included. Many charge extra meaning you end up with an unpredictable/ higher bill (eg £999 for supplementary embryology per round at Lister + up to £800 per round for bloodwork) 2) if you have fewer than 4 embryos for testing you can be partly refunded or roll the untaken spots over to the next cycle. 3) one frozen transfer is included in the package - Lister, for example, doesn't include one in theirs. Additional embyo transfers are £1600 vs £2350 at Lister. Picking on Lister here but just one example of how the clinic price differences can really start to stack up.
You mention your test results being ok - can you remember what they were? Knowing your AMH and AFC may help people to recommend clinics more specific to your needs. If you don't intend to batch, take a look at Access packages. That would bring the price down a lot somewhere like Lister.
If you are likely to produce several embryos I would recommend PGTA testing as others have, if you can afford it. This is to save time and heartache. Just 15-20% of eggs at 42 are likely to be chromosomally normal - this variously manifests as non-implantation, recurrent miscarriage and possibly bad NIPT news in the second trimester. The faster you can find a euploid embryo, the better. If you manage to find one, you eliminate most of the age disadvantage and success odds rocket to 50-70% per transfer. Get three, and you have a 95% chance of a baby.
Good luck!