Children below 13 cannot work (there are special rules for TV and theatre work, where a permit is needed).
During term time, children can work a maximum of 12 hours a week, made up of:
up to 2 hours on school days and Sundays; and
up to 5 hours for 13 to 14-year-olds, or 8 hours for 15 to 16-year-olds on Saturdays.
During school holidays:
13 to 14-year-olds can work a maximum of 25 hours a week, made up of up to 5 hours on weekdays and Saturdays and up to 2 hours on Sundays; and
15 to 16-year-olds can work a maximum of 35 hours a week, made up of up to 8 hours on weekdays and Saturdays and up to 2 hours on Sundays.
Children must have at least two weeks free from school and work. And they must not work more than four hours in any day without a break.
There is more detailed guidance on the specific legal limits on the hours children are allowed to work available at https://www.gov.uk/child-employment/restrictions-on-child-employment.
Local Authorities have powers to vary these hours and conditions, so approach your local authority (e.g. City or Borough Council) for details of the rules in your area.