You can apply for an EHCNA yourself. Don't wait for the school to do it. IPSEA have a model letter you can use.
Medical needs EOTAS tuition has to be the equivalent of full time, which doesn't mean school hours because, for example, when you are working 1:1 you cover the content quicker.
The school should refer on to the LA or at least loads with DD's medical team, but things will move much quicker if you prompt the school and LA in to action.
Lots of LAs have a centre(s) where those who are able to can go for tuition. In some LAs they are called PRUs but in others they aren't. For example, in Northamptonshire they have 2 - 1 Northampton and 1 on Kettering. And Kent have several.
Whatever you do, do not deregister. It is much easier to get support when on a school's roll - crudely, you are someone's problem, whereas it is all too easy to forget about pupils EHE.
Whilst evidence is required the LA must not delay arranging provision by insisting on a certain piece of evidence, for example a consultant letter, if it is not quickly available. The LA should look at what other evidence is available, for example from the GP.
alwaysscared you should pressure the LA to provide medical needs tuition. Alongside this you should apply for an EHCP.