I suggest you go and see your GP first. He can run some initial blood tests for you and you should also test your DH and his swimmers. considering you already have two, there would be nothing to suggest something is wrong. 9 months feels a long time (I know, Ive been trying for that long myself) but I can't see them getting worried until you've tried for couple of years because of your background.
If they do find something from the initial tests you can still have further tests on NHS, but you are correct in thinking that you would not get treatment. Private clinics accept tests done in NHS provided they have been completed within 6 months.
There are plenty of clinics in London. Treatment is expensive, an IVF cycle can cost £8-10,000, so you need to make sure you choose one that you feel comfortable with. You can find a list and comparisons on HFEA website. Some clinics also run open evenings, so you can visit before you commit.
So, see your GP first. Have the initial bloods (day 2/3 and day 18-21 of your cycle) and semen analysis completed and then discuss other options.
In the meanwhile, have plenty of SWi and keep it fun. You can improve your fertile CM by drinking grapefruit juice (not from concentrate), taking evening primrose oil from CD1 to ovulation and even cough medicine - Robitussin. Don't take others. It's the active ingredient that helps. And you can use sperm friendly lubricants. There are now a few around, although pre-seed is still the most popular. You can get it online.
Hope that helps. Good luck with your journey!