A quick google search and I couldn't find anything, but speaking from (not very extensive) experience. Moving through uncompacted snow is VERY difficult without footwear. And snow in the countryside is almost always uncompacted.
I would discount alpine ski's since they are very specialised and didn't really appear until skiing became recreational. So for skis think Nordic skiing, even if the chase is based in mountainous terrain.
Ski's over a big speed advantage on a downhill, even a gentle one, if you know what you are doing, they aren't much of a problem going uphill. But become a liability in closed terrain, dense forest, rocks protruding through the snow, etc. The ski's are also a problem if you get into combat. The quick release mechanism most people know of didn't exist until the 60's (I think), so for D&D once they're on, they're on. Getting up with ski's on is also a bit of a problem.
Snow shoes are really just specialist footwear that negate the problems of trying to walk on soft snow.
So maybe something like this:
With no appropriate footwear: Movement is reduced to 5ft and you have to pass a CON check to avoid fatigue after each turn, (trust me, this is probably generous) but cap this at one or two levels of fatigue.
Ski's negate the penalties and allow a free dash action as part of your move, if the whole move is downhill and in a straight line. Actions in melee are at disadvantage. Standing from prone costs all your movement.
Snow shoes negate the penalties, but offer no other benefits or penalties.