just call the lea and ask who the sta for hearing impaired pupils is. (or there might be a list on tinternet - there will be one for VI, one for PD etc etc)
school nurse is the same - no-one ever knows which nurse is allocated to their school, but if you call the school and ask, they should be able to tell you who it is (or the lea will know, or be able to point you in the right direction). some are great, and some are worse than useless, but worth a try.
it doesn't matter where the paed is, tbh. if they are the specialist dealing with that particular child developmentally as well as for a particular condition (they may also have a local developmental paed) they can write to the lea concerned advising on appropriate settings - for example, in cases where statutory assessment was taking place, this would happen routinely. does the child have an existing statement? (there would have been some initial discussion as to settings prior to being offered a special school place)
it may also be worth getting advice from charities such as NDCS in support of working towards speech rather than continuing to emphasise sign.
it's not really my area tbh, but that's where i'd be going. (dd2 has a physical disability and doesn't require meds - we had a small issue wrt swallowing/ choking, but managed to get all supervising staff trained and she has always been mainstream in school - joint placement with special school for nursery)