Quote:
"A grandparent can open a savings account for their grandchild in the child’s name as long as they have documentation, such as the child’s birth certificate".
Source:
http://www.saga.co.uk/magazine/money/personal-finance/giving/how-to-save-money-for-your-grandchild
Best savings accounts for children:
http://www.theguardian.com/money/2016/jan/08/best-childrens-savings-account-interest-rates
Note that most banks require a child to be 7 years old before they can have a savings account.
Banks usually require a child to be 11 years old before they can hold a current account. The ID required to open such an account must show the same address as that which will be used as a communication address for the child, which obviously makes it easier for a resident parent to open the account than for anyone else to do so. However it doesn't actually seem to prevent a grandparent (or, say, a parent who doesn't live at the same address as the child) from helping the child open an account (as it's nominally the child who applies for the account anyway).