In reality, OP, the police would tell the whole lot of you to fuck off. I'm deeply sceptical as to if any of this happened outside your imagination, as it all has the whiff of l'esprit d'escalier, but on the offchance...
If there's one thing the police really get annoyed by, it's people attempting to enforce the law themselves (for, in this case, a non-arrestable offence, too) and in the process stir up breach of the peace, possible assault and the rest. They really, really don't like vigilantes.
The driver may or may not get ticketed. Given the exemptions to the law for exigent circumstances, and that the police would be pissed off at having their time wasted, the chances are they wouldn't. And aside from anything else, until the car is driven off the private land of the carpark and moves onto the public highway, no offence has been committed, and there is no such offence as "intent to commit a moving vehicle offence". If the police nicked someone in a carpark even an intern at your local solicitors' would feel confident it getting it set aside.
You would certainly get an informal "stop being an arse" from any policeman who attended and, if you continued to screech and jump up and down, they'd arrest you on general principles (breach of the peace, obstruction, that sort of thing) chuck you in a cell for an hour to calm down. In this new Chris Grayling world, you'd also risk a caution.
Depending on your job, an arrest under those circumstances would show up on an extended CRB. A caution would show up on any CRB. That you were fighting the good fight about child seats at the time would be a matter for you discuss at interview.
Outside the world of US Police Procedurals. "pressing charges" has no meaning in English law. Charges are brought by the CPS, and in this circumstances they would be too busy pissing themselves laughing to even read the paperwork.