Here's a hand to hold Beetle
What the GP has advised for you is the standard first step in fertility investigations, so that's good. It's progress.
As for not being eligible for IVF on the NHS. Me neither. Because my DP has had a vasectomy, then got it reversed, and now has better than average sperm count.
I have never been in hospital, never cost the NHS anything, and am not to blame for the poor choice my DP made with his first partner. In actual fact, he's saved the state money by not conceiving any unwanted children with her in the first place. And I actually blame the NHS for doing it, what GP would perform a vasectomy on a 25 year old unmarried man with no kids? But if we struggle to conceive, with his 'fantastic' sperm count, so it would almost certainly be down to my own issues - then no, we're not eligible either.
But if DP had killed his sperm by smoking, over eating, and drinking alcohol, we would be eligible. Fantastic.
On the plus side - the waiting lists for IVF can be quite long, so at least if you know you would have to go private, you can get it done quite quickly and call the shots about when and where.
But don't write yourself off yet - chin up love, you'll feel better later.