I would think that, even though the pupils don't pay, the operation would be classed as 'hire or reward' as the operator gets paid (and, indeed, pays his son).

I have observed coaches collecting pupils from schools, and noticed that none of the drivers were wearing seatbelts.

As I understand it, only stage operations are exempt
(though perhaps it could be said that dropping pupils at different locations is stage work?).
Maybe it was private hire as Taxi drivers are exempt - but is there a stipulation on what size the 'taxi' must be?

Well in sweden it is a bus if it has seats for more than 8 passengers. But it can still be a taxi service, but the driver must have D on his license.

I was once pulled by the Polis when I was driving Taxi and was told off for not wearing my seatbelt. I should have been wearing it as there were no passengers in the car. I didn't have to wear it when I had passengers in the car.

You don't need to wear a seat belt if you're:-

a licensed taxi driver who is 'plying for hire' or carrying passengers

You always think that european countries have the same rules but then you soon realise that thats not the case.
In sweden all drivers must have seat belts if the vehicle itself has it.
All vehicles, except veteran vehicles and some local buses, must have seat belts.
If a bus have seat belts for passengers it is the drivers responsibility that people under the age of 15 uses them. People over 15 may get a fine if not wearing it, but the police must catch them while the bus is running (it is not illegal to not wear a seat belt on a bus that is standing still)

The stage buses do not have seatbelts fitted, whereas the 'coaches' which ply the same routes are fitted with seatbelts and have signs stating that seatbelts must be worn, but nobody bothers with them.

A bus is best described (by me) as having a low floor and with space for pushchairs and wheelchairs, with double folding doors, whereas a coach has standard (high) floor and a 'plug' door and lockers underneath for luggage accessible from outside only.

Bus - Mercedes Citaro or Volvo B10BLE Wrightbus Renown
Coach - DAF SB3000 Plaxton Prima

The coaches require tachographs whereas the buses don't (though they seem to be fitted). This is because the 'route' is less than 50 km (31 miles). The 'coach' route extends to 40 miles (65 km).

When operating over common sections of the route the stops are the same for both buses and coaches, however, the coaches tend to travel between settlements (towns and villages) using major roads whereas the buses tend to stick to local roads.

In sweden both buses and coaches have tachometers. We have bus routes that are longer than 50 km.
Both coaches and buses may, or may not have seat belts, but it is more common with buses without seat belts for the driver. For passengers there are coaches with no belts as well. And people dont bother here either, but it is their responsibility. If they're over 15.