I'm so sorry to hear about your loss Harriet
I have been through 4 cycles of IVF. We never conceived naturally (we had ICSI due to DH's sperm count in single figures), but we conceived on our first and second IVF cycles, which resulted in a miscarriage at 8 weeks and an ectopic/miscarriage ( a nightmare because we had two embryos put back and one developed in my tubes and the other in the regular place and we lost them both). The third cycle failed, but I am currently 12 weeks pregnant after 4th cycle, so I know the pain of conceiving after years of trying and then losing the baby.
The IVF clinic should always make you wait three natural periods before beginning the next treatment after a miscarriage, so it will depend on when your mentrual cycle settles down and the pregnancy hormone leaves your system. If you started the IVF referral process now, the soonest that you could begin down-regging drugs is after your third period.
Make an appointment at your GP for a referral. You don't say how old you are or what fertility issues (if any) meant that it took you seven years to conceive, but I would say that if you are much over 33, I would go definitely private. The NHS waiting lists are a nightmare, although we did wait and had two cycles on the NHS (and then 2 private) because that was what our PCT provided. I was 31 when we went to the GP, I'm 35 now, so it ain't a quick process. In London, a private cycle is about £5k depending on drugs prescribed and clinic chosen.
You and your partner will need to have several fertility tests before going to the main IVF clinic (cycle monitoring and checking FSH hormones etc) which take a month or two. We were at Hammersmith IVF unit, which gives you an all-inclusive price, which was a godsend when I had the ectopic and I had to have blood tests every 48 hours for two months and lots of extra scans.
best of luck