It really depends on the child.
My daughter was 2 and a quarter when she was daytime potty trained (she started with a potty and later progressed to a toilet), and 3 by the time she was night time trained, but she has an overactive bladder still and incontinence problems (she doesn't feel a build up, she just INSTANTLY needs to go, so if she's not within maybe 5 steps of a toilet, she has a small accident), so she wears Tena pads. She's gonna be 9 in September.
My eldest son was fully dry day and night by a couple of weeks before he turned 3. He never used a potty, he just went straight onto the toilet (using a toilet trainer seat). It took about 3 weeks for the accidents to fully stop. He's gonna be 7 in August.
My third child (male) started potty training when he was about 2 years 9 months. He resisted it at first because we tried him straight onto the toilet, and he was scared of it, so we started using a potty and then he was fine.
It took about a fortnight and he was dry both daytime and nighttime. When he was about 3 he switched from potty to toilet.
He's gonna be 5 in September.
My fourth child (male) hasn't started toilet/potty training yet. He'll be 1 tomorrow.
I wouldn't personally recommend pull-ups at all. I tried them before and I found that the child just treats them like nappies, and it's pretty counter productive. If you're out and about, make sure the child uses the potty/toilet before leaving the house, and if they are using a potty, take one with you. I actually used to cart a potty around in a rucksack with me everywhere we went whilst my kids were using one.
Unless there's a medical problem, I think that it's a good idea to aim for children to be toilet or potty trained before they start nursery (which, for kids getting the 15 hours free funding, is the first term after their 3rd birthday). So certainly by about 3 and a quarter is a good rule of thumb. The last thing you want is to be having to take nappies to nursery, and have some nursery worker dealing with your child in that way.
If parents are sending their children to primary school without them being toilet trained, it really does scream of lazy parenting.