Sorry for your losses. It's hard waiting for appointments.
I have also have 3 mc, since last September, and have been seen at my local NHS clinic and privately.
It'd be worth phoning up the hospital to ask what tests they will do and what to expect at your appointment. There seems to be a lot of variation between hospitals. In some areas, you may have a choice about where to go and it'd be worth some investigation as some clinics specialise in recurrent miscarriage.
I had to bring my partner to mine as he needed a blood test (they test both partners for genetic abnormalities), but the letter had not mentioned this and if I hadn't called in advance we would have missed the opportunity to have them done. It took several calls to get through to the right place, and perseverance, but was worth it for us.
I understand that some hospitals ask lots of questions about the losses, which can be upsetting. I did some notes which I handed to the doctor, which saved me having to do this.
At my appointment they did a vaginal scan to check my ovaries, and took blood for testing for lots of things. They said that there would be no follow up appointment unless any of the tests were negative. Some come back quickly, but the genetic ones take up to eight weeks as they have to do something special with them.
I also booked a private appointment as there was a test I wanted to have that the NHS doctor wouldn't do. I've also had a few more bloods done there and am waiting for the other tests.