First off, he was cautioned as soon as he got into the car. This is good practice and correct.
However, the fact that someone has been cautioned it doesn't mean they have been reported for an offence (which is I think what you were getting at?). BUT because of the warning formula wording, sometimes people are unsure they have actually be reported for it. Although there is a stated case, the outcome isn't what is reported online, the onus is always on the defendant to prove otherwise.
The is no direct process, as long as certain things are said and recorded. The fact that the car had video suggests interior recording including audio also.
The reporting and or cautioning should NEVER be done at the end, as everything in front of that then becomes inadmisible, and also an unlawful interview under pace guidelines.
He was correctly cautioned, no paperwork was given as its a summons, and even he the warning formula was not given at the time, it could be forwarded within the 2 weeks (as you said mate).
P.s sorry for saying 'wrong' and buggering off, that was fairly rude, but I had to jump in the bath lol...