@Ella do you have any link for private clinics?
I would do a Google search of local consultants. I found it was individuals rather than national private services
Our health visitor also gave us names of tongue tie consultants who other people had used happily (though was at pains to say they weren't recommending). Might be worth asking them
I ended up just taking my little one to a private specialist as kept getting dismissed. I’m local to you, message if you want details.
Our health visitor, not GP, was the one who referred our son to the tongue tie clinic. Our local children's centres have bookable infant feeding assessments/advice, and that's where we had a feeding specialist health visitor examine his tongue tie. I think in the UK though tongue ties are only treated if they are impairing feeding. It's worth having a look at the signs of bottle feeding challenges too (e.g. dribbling, gassiness, losing latch easily). As to get it done on the NHS they would be looking for that. Otherwise you might need to look at a private tongue tie consultant.