Did the school know about her tonsillitis at the time of her audition? If so they would have already factored this into their decision and looked to see if she was extremely strong in dance and acting and they would have been able to make an assessment on her voice, even if it wasn't at it's strongest, and maybe put her on the reserve list. The fact she didn't make the reserve list doesn't bode well though.
If not then I'm not sure whether they would accept an appeal. The world of musical theatre is beyond competitive - I watched my sister go through it. You have to be the best of the best to get these places at 16, or parents with deep enough pockets to fund somewhere like Tring Park.
If it had been an audition for a role in a show then she wouldn't have got it - there are always other people who can take your place. They can afford to pick and choose. This might be a bit of a wake up as to just how good you need to be (sorry if that sounds harsh).
All is not lost though. There are amazing musical theatre courses available at 18 which you can also get funding for. She has two years to work on her triple threat skills and make sure they are all as strong as each other.
I know one twin who got into the Brit School, albeit on a different stream, and the other didn't despite them appearing professionally on stage and TV regularly. It just came down to how they performed on the day.
Sorry if I sound a bit down about this but it is such a tough world to be in. I am already managing my own daughter's expectations as she adores musical theatre but lacks the skills in one area. It's just not going to happen regardless of how much she wants it and its my job to guide her through this.
Fingers crossed though!!!