1. What second flight?
The student was briefed to fly three legs, from A to B then C and back to A.
He did not land at B but diverted to C therefore, he has deviated from the briefed and authorised route. Before continuing, he should have been briefed to contact the briefing instructor before taking off again on the second flight. (Otherwise, the authorisation sheet /Tech Log will be incorrect). As the weather was clearly not as expected, the instructor should check that the weather is suitable for the return flight.
2. The fog was forecast to clear during the day!
That implies there was fog when he departed! Fog has a habit of not clearing! What published minima does the school use for permitting such flights to depart? They appear to be inadequate.
3. The only recognised PPL syllabus in the UK that complies with the JAA requirement is the AOPA syllabus. That includes student briefing certificates for all solo cross country exercises and a specific briefing certificate for the Solo X-Ctry flight. The AOPA syllabus lists all of the the items that the instructor should brief the student on, prior to the flight. If that had been followed this situation would not have arisen. The AOPA X-Ctry certificate is submitted with the licence application.