Our ds was completely non verbal upon entry to reception. We had, by then started private speech therapy. And it was worth every penny.
Depending upon your local authority, you may or may not be offered help. At her young age, they may not be so concerned.
However, I would def recommend a private assessment. The facilities they have to help them reach a diagnosis should be far better than your LA. We had an LA assessment of sorts and they gave our ds a broken dolly and 2 blunt pencils to play with. The private practitioner had a vast selection of toys. Not to mention more time. They should be able to write up a report, at a cost I'm afraid, which you can then take to your GP/ nursery/ school.
SLT have really suffered financial cut backs, so be aware, it may be hard to get anything at all. We were offered nothing, so had to dig deep.
I would get cracking before reception, because I fear any school will be on your case pronto of your child is not talking. Speech, comprehension, and listening skills all come under the same umbrella so far as our experience is concerned.
I don't want to scare you, and I do appreciate financial constraints. it may be a very simple and relatively easy problem to fix. Only a good speech therapist will be able to help.
Our reception year was hell, and I never ever want to repeat that. They slapped my son with SEN. It was awful. However now, we no longer have private help, he gets once a week slt at school, thanks to his Sen.
There is no way our fabulous ds would be where he is now without the intervention we sadly had to pay for. He is still often struggles with complex conversations that aren't under his control, but we are enormously proud of him.
Look to see if there are any free or low cost chatterbox toddler mornings you may be able to attend. They might guide you.
Good luck, I hope all will be well.