How old are the children, and what are the savings for? You don't have to answer obviously, just something to think about!
Are you a parent, grandparents, or someone else? This might well determine the type of account you can open on their behalf.
For instance, are you a parent saving into a Junior ISA? If so - are you happy for your children to take control of it at 16, and have complete access and ownership of the money at 18 years old?
If so, and the money can be invested for at least 5 years - (and you don't need to access the money before the child is 18) you might benefit from a Stocks and Shares Junior ISA - such as the Vanguard Lifestrategy funds ISA, which is easy to manage and has very low fees.
In the same circumstances, but where you are not the parent; or you don't want your children to access the money at 18 - or indeed you want some say over what they might spend it on (such as driving lessons, house deposit) you might consider the Vanguard ISA fund, but opened in your own name and which you manage on their behalf.
At the moment these routes of investing are paying much higher rates than cash savings - but of course, carry higher risk.
If your children are sensible with money and you want to teach them money management, you might consider a children's current account that pays interest - such as Halifax Money Smart account which pays 1% on balances up to £1000.
Are you a relative looking to leave a longterm legacy for a child? You could consider investing in a pension for them, For every £80 you invest the government would top up another £20 in tax relief.
Vanguard don't offer this at the moment but an Aviva Stakeholder pension can be opened for a child of any age, with minimum contributions of £16 a month (made up to £20 by the government), but you can stop and restart contributions at any time. Fees are capped on stakeholder pensions and you have freedom to move the investment to another provider if you wish.