OP, your DC is going to do well in life as they have a parent concerned about them, and wanting to know how to help.
My DC1 of 3 was a really slow developer: saw a speech therapist for a bit aged 3/4, not particularly interested in readying. On his first day of primary school, when the teacher called the children over the carpet and asked parents to wave goodbye, he promptly sat down on the carpet, facing the ‘wrong way’ (ie everyone else sat down facing the teacher and he sat down looking at the back wall.
If I’d known about face-palming 12 years ago, I would have done it!
He really took off in year 4 - got into grammar school- and was predicted 8 9s and an 8 before - well, you know. And his speech and language is fine 
I am not saying all this as a stealth boast, but to sympathise. Of my 3 children, I have spent 80% of my worries and grey hairs over the first 11 years of his life - he really struggled socially, and with life generally really.
But since starting secondary, he has just... grown into the person he is meant to be. Grown comfortable and happy in his own skin. It’s the loveliest thing to see, and I am so so so grateful. And I wish I had known this 14 years ago, and it would have saved his father and I a lot of angst.
So - in short - I sympathise, but as someone almost out the other side, and works in a primary school with lots of children, they really do often flourish in their own time. It’s stressful when everyone else seems to be on the white book band and yours in still on red or pink (kipper and biff anyone?), but it does often all shake out in the end.