I just wanted to post what appears to be working, a few weeks in and after trying several things, in case anyone finds this in future and wonders how we got on.
Trying to get him to drink a full cup was a nightmare - would only take sips, would complain, cry, we'd find ourselves getting worked up and he'd end up drinking about 2 sips before we'd abandon it so we didn't shout.
Juice/ice/warm drinks etc - interesting for the first drink but then no different to just drinking water. So we've abandoned that too unless he specifically requests something.
What seems to be working is letting him choose which colour cup to have (we have Ikea plastic beakers which come in 6 colours and are a big hit) and then only half filling it with water, so about 150ml. He won't drink it without prompting - usually still untouched at the end of a meal - but when we prompt and say he can't get down until it's finished he will drink it all without much fuss. Similarly at snack time, or when I think it's a while since he's had a drink, I give him a drink and say he can't carry on playing until it's drunk and/or we'll do something fun once he's had his drink. If he doesn't want to drink, we then offer a straw which helps make it a bit more fun, and even let him blow bubbles in the drink provided he alternates between sips and bubbles!
In addition, I'm making sure he has food with high water content like satsumas, cucumber, berries, yoghurt, ice lollies. Going to make jelly for tea tonight as he loves that. We also had a surprise hit on holiday last week with a cinnamony milk drink, so I think milkshake/smoothies might be popular.
He's getting about 5 drinks in him a day, so about 750ml in total, and at least he seems to now accept that he has to even if he doesn't particularly want to.
The result is that his wee is usually pale yellow and not smelly, and he's having about 4 or 5 wees a day with maybe only one accident late on when he's tired. Such a huge improvement on where we were before Christmas - yellow, stinky, about 2 wees a day and lots of small accidents, probably from an irritated bladder.
The constipation is also a lot better - still have small soiling incidents but nothing major but he is getting to the toilet now and he is actually pooing properly. I think his upped liquid intake is really helping.
Thank you all for your tips and support. This thread has been so very useful to me and DH, and DS too of course, even though he doesn't know or appreciate it! :o