I'm in North Wales and am a single Mum to be (currently 32 weeks pregnant with donor sperm at age 40). I unfortunately had to do 6 rounds of IUI to prove infertility to get on the waiting list, but then feel very privileged/lucky to have had 2 cycles of NHS IVF with multiple transfers. It's been about a 3 year treatment process for me to finally conceive once I started, but you could well be luckier.
The problem is no one can tell you how long the wait will be as it varies so much from clinic to clinic on the NHS. Plus clinics only have a certain amount of NHS funding so sometimes even if you have started at a clinic, their NHS funding pot will run out and you might have to wait several months to actually start the treatment (as happened to me).
However sometimes once you are actually in the "process" at the clinic, (tests being done, mandatory counselling arranged, IVF protocol discussed, sperm donor found etc) the opposite happens and they may try to expedite you if they are able - since in theory if you meet the criteria you should be entitled to 2 cycles of NHS IVF before the age of 40 in Wales (and you will struggle to get both cycles if needed before the deadline otherwise). This also happened to me - I seemed to wait for ages and then they expedited due to my age once the funding was there and it all happened more quickly.
Oh and on the private tests front - clinics will usually prefer to do their own scans and blood tests (especially since blood test results are often only "valid" for 6 months). Tests can usually be done quite quickly - it's the fitting you in for actual IVF (multiple follicle growth scans, egg collection, any freezing, embryo implantation etc) that's the issue as they will only be able to take so many patients at once and they don't want to risk you getting "held up" by too many other patients once you have actually started the drugs.
Hope this helps and good luck!'