Righto.
You can get private speech therapy assessments, but to be honest there isn't a great deal they will tell you at this age that you don't already know.
She is still only 2.8. It's young.
the reason (which I have only understood recently) they ask you to come back at three is that she won't be able to sit through a lot of the groups and stuff for speech therapy, she won't understand the games they play. They will struggle to assess her speech and understanding because her concentration is still toddler level. HOWEVER THAT SAID an appointment can take a long time to come through, and I think for things like that you can INSIST that your child is referred, which I did.
They will make you feel like you have Munchausen's Syndrome by Proxy for going against their 'advice' but I followed my instincts - ds1 still has speech therapy now, at 4.5, so they don't know everything.
On the bright side, there are LOTS of things you can do.
Do lots and lots of labelling with her, bubble blowing, drinking with straws, lots of taking turns games, and a good idea would be to get hold of some jolly phonics of the internet, and teach her the sounds and actions (don't bother with the letters, she's too young) as the sounds are all speech sounds, and will help her to clarify.
NHS provision isn't greaT. But it sounds like she is making headway, and in a way, frustration is good in that it will drive her to really try.
for her right now though./