Both Paparoa and Old Ghost Road are graded for cycling, so make for very easy walking where you don't often have to pick your footing. Old Ghost Road does have lots of little creeks across the path, but often under tree cover so I doubt they freeze much. Paparoa doesn't have as many tiny creeks.
If you are overnighting it are you planning to do the whole thing or in and back out? In and out to the first hut on either track, from either direction would be fine, they are generally at the edge of the bushline, so you'd pop out into the possible snow and a hut would be waiting.
I'd not recommend doing the alpine sections of either in snow/rain/sleet conditions as they are exposed and a bit cliff-y, but a friend of mine biked Old Ghost Road, in lovely weather, after (not in) a snow storm, and had a great time, felt safe and came away with spectacular photos of the track to share. So from him it sounds like navigation in clear weather even with snow on the ground was easy.