I've been asked more and more if I test children, and that's probably due to NHS assessment delays.
The short answer is yes. I am trained, confident and willing to test and examine children and teens. I have a good deal of experience of autism and I prefer to do home visits for younger children (and all children with autism), to keep them in a relaxed frame of mind.
Children of school age can generally be relied on to sit for a traditional hearing test (challenging but fun, as a practitioner), but those who are younger need a specialist test in a sound field (room with calibrated open-air speakers), which I don't have. I don't test pre-schoolers for this reason.
I provide test results and reporting for schools/carers on the immediate needs of a child who has abnormal test results. I also provide written referrals to medical professionals, however you should not need this to get NHS medical support for your child. It may be useful however, if your GP has indicated that "they don't do hearing tests for children any more", which seems to be shorthand for "I think this problem is minor or will go away". The NHS absolutely do do hearing tests for children, but they only do so by referral from a GP.
If you are looking for someone to give you a yes/no answer as to whether your child has a hearing loss, and advice on how to deal with that, then I can absolutely help. I can also diagnose tinnitus and provide advice on how to help your child, although my full tinnitus consultation is only appropriate for older teens/adults.