If a member of the public was able to give evidence that they saw somebody famous like Jeremy Clarkson driving his car and that he was using a mobile phone (or committing any other offence) , then that would be enough to prove the case (in the sense of being sufficient to secure a conviction if not disproved.)
The fact of the car being driven on the motorway could be proved by a credible witness saying 'At (time day and date) I saw such and such a car being driven on the M XX towards Roadhogville.' Any photo would just tend to corroborate the identity of the driver. (Obviously, if it got as far as a disputed hearing, the defence would try to disprove that evidence.
Whether the police would investigate that sort of allegation is a bit of a different matter.
A case which received a lot of publicity at the time (30 odd years ago) had no photographs, but involved a public figure, Sir Gerald Nabarro MP. He was one of those people who are sometimes described as 'colourful'. An absolutely stereotypical Tory MP: booming voice, handlebar moustache, bowler hat and so on. He had several big cars, possibly Rollers, with the plates NAB 1, NAB 2 etc, long before that sort of thing was fashionable and he was something of a motorists' champion.
I cannot now remember all the details but a member of the public (possibly several) complained that they had seen this well-known and immediately recognisable character driving the memorably plated NAB 1 the wrong way around a roundabout. In the face of a lot of booming protests he was prosecuted. The defence was that his female secretary had been driving. The court preferred the evidence of the prosecution witnesses and he was convicted. More booming protests and another member of the public remembered seeing the car being driven by a woman and came forward. A retrial was ordered and this time the case was dismissed to more booming from Nabb. I think he died fairly shortly after being acquitted.
I mention this case just to show what can happen. I'm not sure such a case would be followed up today.
Since posting the above, Ive found
this I had forgotten the case involved an accident after the roundabout incident and that report makes no mention of the independent witness coming forward. I was unaware of the Hamilton connection.