1 - We started the NHS round first but as I was at high risk of OHSS (www.mayoclinic.org/diseases-conditions/ovarian-hyperstimulation-syndrome-ohss/symptoms-causes/syc-20354697) they had to freeze my embryos, so we took the opportunity to do a private round at this point, before transfer and while our initial embryos were on ice, to collect more eggs to "bank". This was totally our choice and a bit of an unusual thing to do but felt like the right decision for us as I am in my mid 30s and we want multiple children if possible. It is also worth noting that they wouldn't have transferred the "private" embryos on the NHS under the 3x transfer within 12 month rule.
If you have low AMH you are probably not at risk of OHSS so they will likely do a fresh transfer in the same month as your egg collection on your NHS cycle - this is the preferred option and on the NHS you don't get to opt for a frozen transfer at a later date unless you medically need to.
2 - We might have been very lucky with availability but for my NHS round the initial consultation was 12th Jan - less than 6 weeks after being referred - and the cycle started as soon as possible after this. I had to wait for my period, go for a scan at day 17 to check I had ovulated and then start medication with my next period which was on 1st March. Egg collection was 14th March. Timings were the same for the second (private) round but I didn't need to do the initial consultation. Essentially first egg collection - wait for period - wait for day 17 for scan to check ovaries - wait for period - start meds - egg collection c.day 14. My second egg collection was on the 7th May.
3 - I had to leave 1-2 months to allow my ovaries to "calm down" before they could transfer an embryo, but if you aren't at risk of OHSS they will do a fresh transfer 3-5 days after egg collection. Mine was a bit complicated as I initially wanted to do a natural frozen transfer cycle but my June/July cycle was extremely long - I think 45-50 days - so by the time I ovulated my womb lining was not optimal so they had to push back for another month. We opted to do a medicated transfer the next month instead which is much more predictable. Eventually we transferred an embryo at the end of August - who is currently 23 weeks and giving me a good kick!
FYI for the second cycle I was enrolled in the PGT-A genetic testing trial they are/were doing at kings which would have also meant they had to freeze the embryos and wait a month, but I would have had to freeze anyway so it was a no brainier.
I would recommend reading the book it starts with the egg - it could be complete coincidence but I started doing the supplements and eating better as it recommends in February, so less than 1 month before my first collection but 3 months before the second and I had much better results second time round. 18 eggs vs 13, twice as many fertilised and we ended up with 5x 5/6 day embryos instead vs just 2 from the first round.
I am happy to answer any questions but I can only tell you my own experience and as your causes are quite different to ours they might have you on completely different protocols. I hope your GP appointment goes well this week and you get a really speedy referral, the whole process is a bit of an ordeal but at least you know you are doing everything to push things forward. Good luck!