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.
I'm sorry, I don't think that follows.
The conditions can be summarised as:-
I) if a child is under five
and the child is accompanied by a fare-paying passenger
then they may travel for free
II) if a child is travelling for free
and there is a free seat
then the child may occupy the seat.
These conditions say nothing about a child who is not travelling for free.
The conditions of service also state 'You are allowed a maximum of one seat reservation per person for each train that forms part of your journey'. Good luck to you if you are going to argue with a member of staff, being harassed about the lack of seats, that your imaginary child needs a seat, they will probably (rightly) say that you are one travelling alone, so need to budge up.
Incidentally, when I booked in advance on South Eastern last week, I was given reservations but not for a particular seat.
My comment on the imaginary child sitting on your lap was intended to be flippant!