Prior to school-leaving age, you can't undertake any form of employment before the age of 13. At this age, working in a kitchen is not specifically illegal but school-age young people may not work in an 'industrial setting' (e.g. in a kitchen which is mass-producing foods, perhaps by pre-packing sandwiches for sale in stores). Also they may not do any work which 'may be harmful to their health, well-being or education'. Nobody of school age can work without a permit from their local education authority and the authority would consider whether there were any specific health & safety issues, which might cause concern, before issuing a permit.
Link to information about employment of school-age young people:
http://www.direct.gov.uk/Parents/FamilyIssuesA ndTheLaw/ChildrensRights/ChildrensRightsArticl es/fs/en?CONTENT_ID=4002945&chk=AaZA0p
Once school-leaving age has been reached, there are no legal barriers (on the basis of age) to anyone working in a kitchen.
Everyone who is involved in the preparation of food must possess a food hygiene qualification. Basically, that means everyone who works in a kitchen except, possibly, someone who, for example, simply does the washing up. (Even so, health inspectors would certainly expect that someone doing the washing up should still be able to demonstrate knowledge of some basic rules of kitchen hygiene).
Chris