Struggling to get doctors to realise my seriousness

Hi all, my daughter will be 3 in April and for a while now we’ve been thinking she may be autistic. She’s unable to speak, she hardly listens to us, hates been around other children and when she is, she doesn’t socialise with them including her sibling. We’ve also noticed that she stims quite regularly and does what we call her ‘excited dance’. There’s loads of other small things that draws us towards autism, but don’t want to list them all.  She’s had several hearing tests which have all come back fine too. We went to see the doctor today to talk about our concerns and he basically said he doesn’t deal with autism and gave us a number to call. So I called them this afternoon and they said there’s nothing they can do unless she’s referred to them by her school or a speech and language therapist. She’s too young for school and we’ve been on the waiting list for her to have speech therapy since October last year. 
Does anyone have any advice on what we can do now? Feel like we’re stuck and just want to get to the bottom off it all. Thanks.