Parsley I think a lot of the testing offered by GPs depends on your area, and how helpful/sympathetic the GP is. I believe they're all supposed to follow a guideline about testing/referrals based on your age, but some GPs take the attitude that 80% of women get pg in the first year, and 95% by year 2 (or something like that) and push you off as much as possible. I had one like this and promptly switched doctors. Basically I found out what tests were offered based on age, and went in to request them as I hit each stage.
If you aren't offered something you feel you need, or if you feel like the wait is something more than you can stand, by all means go for a private MOT. We did, mostly to see if my tubes were blocked, but also bc we knew NHS IVF was never an option due to DH's existing children. There are plenty of places that offer testing packages, and you could then take results to your NHS FC appt, if your GP refers you, which could save you time.
Ultimately the tests DH & I had over 20 months were: FSH/LH/Progesterone (two sets), SA (twice), pelvic ultrasound, HyCoSy (privately), AMH (privately), antral follicle count (privately - all at same appt by the way), Clomid x3 (only took 1 month though, but it was monitored, and one of the scans was what revealed a polyp), subsequent polypectomy (privately but NHS would've done it if I'd opted to wait 3-6 months), and then all the bloodwork required by the IVF clinic we chose - full thyroid panel (more comprehensive than just a standard thyroid test), rubella, chlamydia, HIV, Hepatitis etc (varies by clinic). There may have been other tests on offer but I had no history of any complicating factors, so these were what were thought appropriate, or could possibly explain our "Unexplained Infertility" (which hopefully as of last week has been proven wrong!)