I work in car insurance claims.
Firstly, you won't be able to do much for at least a few days so your best bet is to concentrate on your ds.
To make a claim, you're going to need the drivers name, registration and insurance company, then it is a case of calling them and seeing if they can help or instructing a solicitor.
I would advise dealing direct if possible, as it will be quicker.
The insurer will need to have spoken to his insurance before they can help you, which is why I suggest you wait a few days.
Regarding the compensation culture - claiming because you can or to punish the driver is what perpetuates the problem. Having a genuine financial loss (like your pram) is not the same thing
Of course this accident shouldn't cost you any money.
For the injury, it's too soon to make any decisions. If you are injured, you may not really feel it for a day or so, and you have time to see how you feel and if you have anything to claim for.
Also, a charge from the police for the driver is completely separate to your claim. You can't push for this and the result of the police investigation will have no baring on the claim.
And lastly, to the poster who said she hopes the driver feels awful, he probably does. No one deliberately runs down a mother and her child - quite often the drivers are traumatised too - I've never spoken to a driver in an accident like this that wasn't deeply effected.
I hope your ds feels better soon OP 