When it comes to a regular bank account, I think my simple answer is "When they needed them"?
I had one when I was 13 because I had 2 large paper rounds (1000+ papers a week) and you needed a bank account for them to pay you.
My daughter got one when she was 15 when she got a reasonable lump of cash as a present and decided she wanted to 'save', so we helped her through it.. She's then used it when she got a weekend job whilst in 6th form.
My son however has been fairly laid back and has no desire, although he's talked about getting an evening or weekend job, so will probably open one shortly just in case..
We have ISA's and other efficient savings vehicles for them, but that's a different story.