I can't speak for the others, but BA allows miles to be earned by fliers under 18 if they are part of a "Household Account"
16.1.12. Parents or guardians may nominate persons under 18 years of age to be a member of a Household Account and this will entitle them to earn and, subject to the remainder of this Clause 16.1.12, redeem Mileage.
Miles accrue to the child's personal FF account.
I can't imagine any airline denying mileage accrual to a minor flying on a paid adult ticket. There are many stories here about five year olds being top-tier elites in a number of programs.