If you’re in Nenthead then the 2 obvious routes are posted above and on the other thread – Cross Fell (ride over the C2C track through the mines past seldom seen to Garrigill or the much steeper road option then you’re away, use the flagstones between Cross and Great Dun Fell) or High Cup Nick (described vaguely by OAP on the other thread you link to) are pretty OK draining and well surfaced for CX bikes. Unless of course the snow comes down or the wind really blows in which case they aren’t places to get stuck at this time of year. Quite long days out for most though, but you can link some good roads in and make quite good pace on a CX bike if you’re up for it. Often at this time of year there’d be better skiing than biking for quite a bit of these routes but as long as you avoid the trout beck singletrack (non existent/fallen into river) then these are probably reliable.
Any of the bridleways further north (as described and linkes above by legolam/ paul4stones) are best avoided unless there’s a big freeze or a miraculous drying up.
You’ll note none of these are actually in Weardale though (but then neither is Nenthead). If you are in Weardale then there are some loops if you mix road and landrover tracks (and like climbs/descents) but nothing amazing. Definitely worth exploring with a CX bike though , if you look at an OS then you should be able to plot loads of possible routes linking from Nenthead – Killhope – Weardale – Rookhope – Weardale – Teesdale – South Tyne using the roads (quiet, good climbs and descents but not great surfaces on the smaller ones) and the byways/ bridleways/ C2C sustrans bits between them (stick to the marked double tracks which are mostly pretty well made mining or estate routes and stony rather than grassy bog or peat hags) and you should be fine. This can be as hard or as long a day out as you like – recommend a stop at the Cowshill Hotel if you’re passing.