I know you haven't had any responses and I'm in the US so I'm not that up to date on how the NHS works with that sort of thing. But I would talk to a midwife at the very least and lay out your concerns. Maybe there's a compromise they can offer-less office visits and more telehealth appointments? Bloodwork done as minimally as possible? And I don't know how it is in the UK but in the US, you can decline any bloodwork/scans you don't want. I would at least do minimal appointments with the midwife or doctor simply so you have somewhat of a connection with them come time for delivery if you do plan to use the NHS for that part but as long as you are healthy and things progress normally, I don't see why it should matter if you go private until the end.
Edited to add that I totally understand your views on not wanting to go to the dr. I have medical PTSD surrounding pregnancy because of a lengthy history of miscarriage and a stillbirth as well as a really REALLY traumatic labor/delivery situation that ended up with a month long NICU stay. This time, I tried to push for holding off on an ultrasound until I was out of the first tri but my dr convinced me to go in because I was potentially carrying twins and some of my labs were showing suspiciously high-for-me hCG levels. I also didn't know my exact dates and I have tiny babies due to suspected growth restriction so I need monitoring to make sure my babies are growing well. But we're talking about my wants/needs for this pregnancy and she's okay with me only having one growth check per trimester, no fetal stress tests unless growth is dropping, and as long as I'm feeling okay, I don't need to be seen any more than necessary until the end. I tend to have my babies early so I'll likely need to go in for weekly appointments starting at 35 weeks but I can deal with that. It's also in my chart that only MY doctor is to deliver my baby and if I want an elective C-section due to the anxiety delivery causes me after my traumatic birth, she's 100000% on board. Having a supportive doctor can make a world of difference.