Not from London but my baby was diagnosed with quite a severe tongue tie after birth. We were referred to get it snipped, it took about 7 weeks for our appointment with the NHS.
Breastfeeding until then was horrendous, particularly on one side where my nipple had basically been destroyed by her.
The infant feeding team and lactation specialists came out multiple times to see me and confirmed I had her latched correctly, it was the tongue tie that was causing the nipple destruction and agony.
After my nipple healed and after her tongue tie was snipped, it was much less painful and then not painful at all after a week or so once she got accustomed to her new range of tongue movement.
In terms of the procedure, I just held her, the doctor put his finger in her mouth and pressed down to make her tongue lift, and got some tiny scissors and just snipped the skin under the tongue in one motion.
She cried obviously and there was some blood. It looked like there was more blood because it mixed with her saliva.
I went into a room immediately after and breastfed her for 15 minutes or so and she was fine. Fell asleep in the car on the way home. No issues at all after
She has refused a bottle and a dummy since the tie was snipped. Before it was snipped she occasionally had a bottle of expressed breast milk whilst my battered nipple healed. She took to the dummy at about 4 weeks old too
Shes nearly 8 months now. Still Breastfeeding. No dummy and no bottle because she still refuses them, but to be honest I'm quite happy about her not being reliant on a dummy anyway.
Good luck! I would say go for it if you think its best. The pain is over and done with in a minute