Hey OP - I've got a 13 yr old who is still in nappies, so we've tried various things over the years.
Is your DC particularly big or small for their age? Or do they have a particularly small waist? My DS is very slim and finding nappies/pullups that fitted him round the waist was a problem because the capacity for holding in waste wasn't sufficient.
You can be referred to the continence service, as others have said. Personally the continence service in our area isn't very good at all (not the quality of the nurses, the products they provide). They're extremely reluctant to provide pull-ups and prefer nappies, and usually they're just brands that you can buy privately anyway. They're not special nappies/pullups as you might hope/expect and don't perform any better. If anything I found them to be poorer quality (presumably due to cost).
Where is it leaking from? Is it through the leg holes because the fit is poor? Or is it because the nappy/pull-up just can't hold enough? If it's the former, you could try putting a pair of regular pants over the top of the pull-up, as that will hold it a bit closer. Or the period/incontinence pants over the top as a PP suggested.
I found the smallest size of adult incontinence wear to be huge - we have to use it for swimwear and it virtually comes up to DS's nipples!
We've found the Huggies DryNites the best - we didn't get on with Pampers at all as DS got bigger. Pampers were great for babies but we found they split.
We also found pullups were better than nappies. I think a PP linked to the ID site. They're quite good and well worth a try - and if you email them first they will send you samples of their different products so you can see if it suits before ordering a batch.
Forgot to add. If there's any abnormalities with your DC's toilet habits - for example, they're weeing/pooing a huge amount in one go rather than going regularly - they can provide help/advice with that too. You might not need this, but just thought I'd mention it.
Sorry there's not a brilliant answer for you. We've been battling this for years and just can't find a failsafe solution. There really is a gap in the market - if there are any manufacturers reading this!!