It's a shame that the government stopped the CTF/Junior Isas - they used to put money in to start them off - as that started a habit of saving right from the beginning.
Both ours have had CTF/Junior ISA from birth and we put a regular small amount in so they're not too disappointed when they hit 18. We prefer to invest most of our money elsewhere though - better financial products, we have more control in case they wanted to blow it all on beer aged 18 etc. They'll benefit from these savings, but won't have direct access.
Once they started getting pocket money aged around 5-6, we opened savings accounts for them, even thought they were only getting tiny amounts, and that increased as grandparents started handing slipping them a tenner here and there, and cash gifts for birthdays etc. HSBC had the best interest + reasonably accessible this summer we found, and Virgin for under 7s, but you need to keep it under review.
At 7 we did a little exercise on bank accounts, interest, compared interest rates across 3 accounts I'd pre-selected as being suitable, then discussed how often they wanted to access the money, and they got to weigh up what they wanted re: accessibility vs. higher interest. Pocket money increased year by year and they got to decide how much to save vs. keep in piggy bank for immediate spending, every month. They also keep an eye on the interest rate changes and decided when to move to a better value account.
At 11 DD got her first current account alongside the savings, and now her pocket money is split across the 2 accounts.
Both (primary aged) are now avid savers who like to read their statements (we taught them), and think carefully about how they use their money - saving/spending/charity, needs vs. wants etc. I think there's huge value in starting financial education young, but obviously we're lucky to be able to save for them.