Warmoth can and will build you a great neck. I've played a few and they're really nice. If you know exactly what you want out of a neck I doubt that that a Warmoth neck will leave anything to be desired. They can build it to look correct for the period to (blocks, binding, ect.) A friend of mine just mated a Warmoth 5 string neck to an MIM jazz body and is beside himself. No complaints. Keep in mind that a Warmoth jazz neck might feel a little chunkier than a factory neck but there is a slim taper option.
As far as affecting the tone I'm sure that it would, but I don't think that it would adversely, especially if it's built using the usual materials, maple and/or rosewood.
I own a Geddy Lee jazz bass and the neck is the best I've ever played on a jazz bass but if you can't find one then Warmoth is certainly a viable option.