I did the HSCT in a day on Monday 15th June. It's pretty much snow free except for a couple of patches in the notch between the Lions, and on the scree slope/boulder field descent from Thomas Peak towards James Peak. Snow was soft with some minor care required to prevent post holing into very uneven ground below but otherwise the trail is in fantastic early season condition. The 2 main snow sections remaining are really short, you won't be on them for much more than a couple of minutes travelling time. The snow patch which remains in the notch between the Lions I would recommend a pair of spikes for. It has quite the banked angle on it, but stepped out enough that there's very little chance of you sliding if you are cautious.
Plenty of water refill points from Magnesia Meadows onwards which don't involve the need to take stagnant, scummy water from a tarn. There's a couple of really nice fast flowing creeks.
If I was to stick my nose out, I'd say go and get it done while there are good water levels and it's not too hot out there. The snow genuinely isn't the problem right now, it's the bugs!! I think by early July it'll be a totally different ball game (although still a bit buggy). Having said that, this is based on my assumption of if you want to go fast and light and not add any extra summits. If you were to add additional summits such as Hanover or Brunswick, I'd hold off as snow persists on the more gentle aspects above 1400-1500m on the more Northerly summits.