My tonsils were removed in the US when I was about 24-years-old (10 years ago). And the funny thing was the waiting room was filled with toddler-sized chairs. Waiting mothers were condoling with my grandparents, who were at the clinic to drive me home afterwards.
I don't know how serious your condition is, but I had such an infection in my throat that as a result, I was afflicted with anemia. I was cold much of the time, and depressed, and generally felt awful.
Sadly, I can't help you with any advice as to what it would take to go through the NHS to get the surgery done. But if you need the surgery, please be persistent. It was one of the best moves I ever made.