DH also has a count of c. 2 million. We waited for over 6 weeks for an NHS andrologist last year which then got cancelled before Christmas. So we saw an andrologist privately for c. 300 GBP. This helped us confirm the issue and gave some (odd) peace of mind over the holidays so I am glad we did. We then had a fertility clinic appointment... and they wanted us to see an NHS andrologist anyway. Took another several weeks for that to be scheduled and have currently been waiting for a month for them to have a follow-up consultation! Just to triple-confirm what we already know, not for any medically useful reason. (We already did all the scans and bloods etc. via our GP last year, so have not had to pay for any of that so far and the bloods especially were easy to schedule via GP.)
Long story short - private is faster and easier but NHS may still make you wait and jump through hoops anyways if you're going the NHS route. It's not like they didn't accept the results, but it's a bureaucracy or admin sort of thing that I am still trying to understand...
We're facing similar issues with NHS scheduling - me - lots of appointments that take ages (even though I don't currently need them it is just to tick boxes in the correct order; the 'issue' is only with DH). So you might want to talk to someone about that as well! Our hospital doesn't even have us registered as 1 case but 2 separate... and all the IVF ICSI comms are going through me (even though DH is why we are having the appointments).
We're still doing the NHS route because we live in a good postcode apparently so it will be worth it. But yeah, also massively impatient, and not sure I necessarily recommend it!
Could just be our experience though. So do see what others say. And also I should add, the hospital keeps telling us that there is no waiting to start the actual treatment once we finally get to that stage.