Hi, Reuben,
As
@SYates already wrote you will most possibly be alone the whole way. And after Segovia it can really be freezing cold because of the wind. It is flat more or less, called Tierra de Campos and actually very much like on the Meseta the winds are strong.
Personally haven't walked it in winter but this year I started second half of April from Madrid (doing just sections but walked whole in 2014 in the early summer) and there was still at least half a meter of snow over the Fuenfria Pass (on the North side) from Cercedilla to Segovia. Also there are no itermediate stops, only three fuentes. If you are not used to this kind of conditions then I suggest taking a train ride from C to S in the winter, it takes approx.40 minutes.
Let us know how it goes, please
Buen Camino!
PS (I don't think you'll have troubles with accommodation. Albergues are more or less locally taking care of. Maybe just call one day in advance. That's also good for your security if went missing in those winter conditions, god forbid.)