Dk can be thick or thin, it's not one thing. If you have a look on the British Knitters board on Ravelry, they have pages where British brands are given which work well as sport-weight yarns. Find the group page, then look in the tabs, they have a few pages on yarn weights.
what they said. Debbie Bliss Baby Cashmerino is sport weight and some indie dyers do them too but it's widely used in us and a bugger to find here thicker 4ply or thinner dk will do. You may want to adjust how many you cast on.
if you look at the ravelry page for the pattern, you can click on the 'projects' tab, and then on 'advanced search', you'll then be able to filter by what other people have used for their projects (wool thickness, needle size) on the left-hand side. according to that, more people have in fact used DK than 5-ply, so it clearly works, but they may have had to adjust the number of cast-on stitches to avoid it being too large depending on their tension
You can also search for sport-weight yarns then look through people's stashes and see what they have for sale. I've got a few American yarns that way and generally people are very reliable. (but it is caveat emptor I suppose)