Many local authorities have byelaws regulating the work of children which can be found online.
NB in some LAs the same person or department deals with employment licences as with elective home education.
The youngest age a child can work part time is 13. See https://www.gov.uk/child-employment
A young person may be “employed” without receiving payment if they are assisting in a trade or occupation carried on for profit such as children who help parents in their shop.
The rules are different for children in the arts, modelling and sports where the audience pays and/or the child is paid. See https://www.gov.uk/child-employment/performance-licences-for-children
A child under school leaving age may only be employed to do ‘light work’ and for a maximum number of hours a day. See https://www.gov.uk/child-employment/restrictions-on-child-employment.
The gov.uk page makes a distinction between term time and school holidays, saying that during term time children can only work a maximum of 12 hours a week, while in the school holidays 13-14 year olds are allowed to work a maximum of 25 hours a week, and 15-16 year olds 35 hours a week.
Home educated children do not have to keep to school days and terms in their education but this does not mean that they can automatically have a job for more hours a day or more hours a week. As explained here in England, local authority maintained schools must open for at least 380 sessions (190 days) during a school year.