All children in the UK have their own personal annual tax allowance. However, anti-avoidance laws prevent this allowance being utilised by parents of children aged under 18 with some minimal exceptions.
If older children are employed by a parent they can receive income paid as wages subject to the usual rules.
There are special rules if a parent gifts significant amounts of money to their children which results in annual bank interest of more than £100 (before tax) accruing to each child. If this is the case, the parent is liable to pay tax on all the interest if it’s above their own Personal Savings Allowance.
The £100 limit does not apply to money given by grandparents, relatives or friends. In addition, any income from CTF’s or Junior ISA’s is exempt from Income Tax and CGT on the child or the parent even where the invested funds came from the child’s parents. The 2019-20 subscription limit for both CTFs and Junior ISAs is £4,368.