Being that your DD has had the complication of a quinsy, which is serious, what is the wait for ENT in your area? Ring the hospital and speak to secretary and explain the problems, how it’s affecting your DD, losing days school etc could take cancellation etc or could you see ENT privately? You can self refer for ENT.
I had recurring tonsillitis and had quinsy twice. I had my tonsils out in my late twenties, the hospital said they should have been taken out at 10, after I recovered from the last quinsy. Your poor little girl OP, I can empathise completely.
My DS1 had many ear infections as a young child, we saw ENT privately in the end. I honestly think that the antibiotics affected the enamel on his teeth, we always seemed to have amoxicillin in the fridge, the effect on teeth is well documented, on one occasion he was on amoxicillin for an ear infection, then he started complaining about the other ear. Rang surgery, to be told no he can’t possibly have another infection in the other ear, as the antibiotics would have covered it, I insisted he was seen, and yes, I was right, he did have another infection, so then we started another course of a difficult antibiotic alongside the Amoxicillin. He grew out of it eventually, but I have never forgotten the awful nights, of him crying in pain, always started in the middle of the night. It made me feel so helpless.
I hope DD is better soon OP.