I did this loop early May last year - here are my observations. Jackson's cone would call for crampons, on the upper 50 feet of elevation, the ledgy part, if it were a sheet of ice. I'd check trail conditions reports, but I expect most of the time this would not be the case. Same would be true for the descent off the north slope, which because it faces north, of course, would hold ice longer than the west slope up which you ascend via the Webster-Jackson Trail, Jackson branch. The first part of that western climb up the cone is also quite tricky, but there's lots of rocks, roots trees, etc., to grab onto and for foot and TP placement, so negotiable.
The A.T./WCT to Mizpah had a well-packed well-defined monorail when I did it, and I gather this is a popular winter lolipop hike, so I'd say, yes, that stretch will likely be well tracked and packed. Note, though, that if there has been recent snow, this stretch, in windy conditions, will certainly have a lot of drifting.
Bottom line - check recent trail conditions reports, but most likely, crampons and ice ax won't be needed, and WCT will likely be tracked. Smart, experienced winter mountaineers tell me, a newbie, that it's smart to bring along crampons, so I'll repeat that advice. I have yet to attempt a winter hike in the Whites, so I'm not yet the voice of experience. Hope you enjoy!
PS: There a couple of rocky, icy stretches of the Webster-Jackson trail before the major stream crossings which could be pretty challenging and icy. Might present some issues.