There is no statutory provision prohibiting a child from owning shares.
Please note contracts cannot be forced against minor thus it’s advisable that that shares are fully paid up.
Dividend income is deemed under ITTOIA/S629 to be that of the parent for tax purposes, and is not treated as the child’s. So there is no tax advantage in holding shares in a child’s name.
I read a good article which gives alternatives to registering child as a shareholder. Click here.
If the client still wants to allot share to a child. Then a bare trust is needed. There is no legal requirement to create a formal document to create a bare trust.
Name in the register of members can be simply – Mr Parent as bare trustee of Mr Child.