Staying moving can help keep you warm. Being stationary for too long is when the cold can start to get to you.
Snow could definitely be a potential hazard when climbing if it’s sticking to the bark or iced over. And even if only on certain parts of the tree, once your boots are wet or have snow on them it can be slippery at times. Just something to be mindful of. If you’re finding the slipperiness of the bark is making the work more hazardous, particularly with limb walking, etc., then it might be better to leave that tree for a day when it’s more dry if possible. Or perhaps you adjust your pace, give yourself more time to move through the canopy. I would also probably suggest factoring this in when setting rigging. Make sure the rope or sling won’t slide off, etc.
Other things to consider is depending how much snow there is you might prefer to clear the work area first. I once had a job where I had to snow blow part of a yard to create access for a lift to trim some hedges. Not only that, but there was ice on the ground as well. Aside from being careful not to slip and fall when walking, I also wanted to be careful to keep the machine / outriggers on parts of the yard that weren’t significantly iced over to prevent sliding and losing traction.
Also, goes without saying that heavy snow and ice can add stress to branches. So when climbing, it’s a good idea to keep that in mind as you’re stepping on branches and limb walking.