This is for Scotland - I don't know if it is different in England:
1. Age of legal capacity
As from the commencement of this Act—
a person under the age of 16 years shall, subject to section 2 below, have no legal capacity to enter into any transaction;
2. Exceptions to general rule
(1) A person under the age of 16 years shall have legal capacity to enter into a transaction—
(a) of a kind commonly entered into by persons of his age and circumstances, and
(b) on terms which are not unreasonable.
(2) A person of or over the age of 12 years shall have testamentary capacity, including legal capacity to exercise by testamentary writing any power of appointment.
(3) A person of or over the age of 12 years shall have legal capacity to consent to the making of an adoption order in relation to him; and accordingly—
(a) for section 12(8 ) (adoption orders) of the [1978 c. 28.] Adoption (Scotland) Act 1978 there shall be substituted the following subsection—
“(8 ) An adoption order shall not be made in relation to a child of or over the age of 12 years unless with the child’s consent; except that, where the court is satisfied that the child is incapable of giving his consent to the making of the order, it may dispense with that consent.”; and
(b) for section 18(8 ) (freeing child for adoption) of that Act there shall be substituted the following subsection—
“(8 ) An order under this section shall not be made in relation to a child of or over the age of 12 years unless with the child’s consent; except that where the court is satisfied that the child is incapable of giving his consent to the making of the order, it may dispense with that consent.”
(4) A person under the age of 16 years shall have legal capacity to consent on his own behalf to any surgical, medical or dental procedure or treatment where, in the opinion of a qualified medical practitioner attending him, he is capable of understanding the nature and possible consequences of the procedure or treatment.
(5) Any transaction—
(a) which a person under the age of 16 years purports to enter into after the commencement of this Act, and
(b) in relation to which that person does not have legal capacity by virtue of this section, shall be void.
A lot of jibberish in there, but I think the general idea is that you can work if you are under 16, so long as it is not dangerous, like working in a mine or something.
But you don't have to be paid the minimum wage if you are under 16.
Hope this helps!