I thought that too, obvioisoy googled it, this id what i found on Rospa.
Exceptions Unexpected Necessity Over a Short DistanceIf an appropriate child restraint is not available, children over 3 years old may travel in the rear using an adult seat belt. This exception is for emergencies and unplanned situations. For example, when someone has to carry a child due to an "unexpected necessity" over a short distance, and where not carrying the child (because a child restraint is not available) might leave him or her at risk. It does not apply to trips that could have reasonably been planned (e.g., a school run), nor to 'long' journeys or to children under 3 years old. You cannot take children under 3 years in a vehicle without a seat belt or the correct child car seat.
Goes on to say
Child Seat and Seat Belt Laws
Children in cars, vans and other goods vehicles must be carried in an appropriate child restraint from birth until either they are 135cm (4'5") tall or 12 years old, whichever comes first. There are very few exceptions. They must then use a seat belt (although it would be preferable to use a booster seat until they are 150cm (5') tall).
Children under 3 yearsMust use an appropriate child restraint in the front or rear of a vehicle. They cannot be carried in a car that does not have seat belts because it is not possible to fit a child restraint in such a vehicle.
It is illegal to carry a child in a rearward-facing child seat in the front if there is an active front passenger airbag. The airbag must be deactivated or the rearward-facing child seat must be placed in the rear.
Children aged 3 years to 135cm tall (approx 4'5") who are under 12 yearsMust use an appropriate child restraint when travelling in the front seat of a car, van or other goods vehicle. They must use an appropriate child restraint when travelling in the rear, if seat belts are fitted.
Children aged 12 years and over (or younger children over 135cm tall)Must wear a seat belt if fitted in the front or rear of a car, van or other goods vehicle.
It is the driver's responsibility to ensure that children under 14 years old use an appropriate child restraint or wear their seat belt.
ExceptionsUnexpected Necessity Over a Short DistanceIf an appropriate child restraint is not available, children over 3 years old may travel in the rear using an adult seat belt. This exception is for emergencies and unplanned situations. For example, when someone has to carry a child due to an "unexpected necessity" over a short distance, and where not carrying the child (because a child restraint is not available) might leave him or her at risk. It does not apply to trips that could have reasonably been planned (e.g., a school run), nor to 'long' journeys or to children under 3 years old. You cannot take children under 3 years in a vehicle without a seat belt or the correct child car seat.
When three child restraints cannot be fitted in the rearIf it is not possible to fit three child restraints in the rear of a car, then two children up to 135 cm in height must travel in child seats and the third child may use an adult seat belt on its own. This exemption does not apply to children under 3 years old.
It may be better for the third child to sit in a child restraint in the front rather than use the adult seat belt in the rear. If there is an active passenger airbag, check with the vehicle manufacturer or the handbook about using a child restraint in the front. If you do, move the front seat as far back as possible on its runners.
A rearward facing baby seat must not be used in the front if there is an active front airbag.
Licensed Taxis (including private hire vehicles and minicabs)If a child restraint is not available, children under 3 years must travel in the rear, but may be unrestrained. Children 3 years and over, up to 135cm tall must sit in the rear and use an adult seat belt. Children aged 12 years or more, or over 135cm tall, may travel the front, but must wear the seat belt. (See the Other Vehicles section).
Emergency VehiclesChildren may be carried in vehicles being used by the police or other emergency services without using child restraints, if appropriate restraints are not available.
Which would suggest, imo, we could have used the taxi without the car seat, but not the tow truck. At the time it was quite stressful and I missed the bit about the taxi. Good to know .
Over aged 3 slightly different rules apply, here's the link if anyone wants a read.
https://www.childcarseats.org.uk/carrying-other-peoples-children/child-seat-and-seat-belt-laws/#:~:text=Children%20under%203%20years,restraint%20in%20such%20a%20vehicle.