Hi Janete,
I've been in recent contact with some walking pilgrims. They started from Le Puy in early November but pulled out when they got to the Pyrenees because of snow - that would have been early December. I imagine there could also be snow in the Montes de Leon after Astorga.
I'm not a cyclist but I imagine that in winter you would be pushing the bike a lot of the time rather than riding it, and I think this would be much more tiring than carrying a pack on your back. There is also the problem of not knowing what is under your feet/wheels if there is snow - rocks, potholes, etc. if you cycle on tracks.
In 2005 I set out on 1st March expecting to be walking into Spring. In Holland I hit 30 cms of snow and the worst March conditions since records began, with night temperatures dropping to minus 10 C. This meant that, for over a week, during the day it was difficult to find a place to rest, it was freezing with nowhere to sit down. I also hit blizzard conditions that year when approaching La Chaise Dieu towards Le Puy.
Winter snow can be beautiful but it is an entirely different matter when it is one of the elements you have to contend with for several hours each day. Even if the snow is cleared from the roads, the snow at the sides can make the road narrower and therefore even more dangerous for the cyclist/walking pilgrim, who may also not be easily seen in poor winter light.
I don't want to discourage you, after all I did survive the above conditions although I wouldn't have chosen them, but there are these things to think about.
Buen Camino
Brendan