Thanks for all the replies everyone!
The OP hasn't mentioned a railcard.
It would be against F and F railcard conditions, obviously, but that's all.
I agree that this would not be allowed with a Family And Friends Railcard - in most cases, it is cheaper for an adult to travel with a child than on their own using one of these!
Would it not be cheaper to buy one ticket in First?
It might be in some cases where a cheap First Class advance ticket is available, however this is unlikely to be the case most of the time.
The ticket entities you to travel and not the seat.
Therefore someone else could sit there should they wish.
And a child has no more right to sit than anyone else - a passenger without a seat might just suggest that the imaginary child sits on your lap.
The National Rail conditions of carriage state "Children under five years of age may accompany fare-paying passengers free of charge, unless the Train Company you want to use specifies otherwise in their notices and other publications. However, children under five years of age who are travelling free may only occupy a seat which is not required by a fare-paying passenger". So someone might buy a ticket for a child under 5 specifically so that they are entitled to a seat, in which case it would be ludicrous for someone to suggest that a child sits on your lap if they have a ticket (out of interest, to anyone's knowledge, has anyone ever been asked to show a ticket for a child under 5 in order to show that they are allowed to occupy a seat? Or has anyone ever purchased a ticket for a child under 5 so that they are entitled to a seat which is not required by a fare-paying passenger?). This condition would appear to suggest that a ticket does entitle you to a seat.
Without going into any detail as this concerns air travel rather than rail travel, I did once see a sign at an airport advising that passengers can purchase the seat next to them if they would like more space.