A friend and I were on the NEB yesterday (June 27). The gullies on the descent from the summit pyramid are now almost completely dry (any remaining snow is easily avoidable).
There was some snow on the way from the scree basin beneath the summit tower to Crossover Ridge (firm enough that we didn't posthole and could kick steps when needed, but soft enough to glissade). Also, there remains a bit of snow beneath the Wooded Stump ("stumpy hill" in McLane's Alpine Select) and on the orange talus field beyond (the photo below gives a vague idea).
If you're banking on finding water along the way: there was snow available near the start of the 4th class section below the summit headwall and at the summit.
Also, a small note: if you're crossing under what remains of the Bypass Glacier, do be careful. It might be more casual than the usual state of affairs, but the remaining ice is still active. A sizeable chunk collapsed as we were crossing the slabs below, and we would have been in its path had it failed a minute or two later.
BTW, since I haven't come across this in a few other trip reports I've seen: There are large cairns adorned with pink flagging tape marking where to leave the talus slope beneath Crossover Peak into the woods, and a flagged trail follows. Eventually, orange reflective markers (nailed to trees) materialize as well. The descent trail is very clearly marked from these cairns, right to the memorial (no bushwhack required; many thanks to the trail-builders!).
YMMV, and all that jazz.