Can't really see the point - the salicylic acid in willow helps with root formation - until they have them (roots), no point in watering and once they have roots, no need to use willow.

I am talking about once the cutting is stuck in say for the first two weeks, before they get roots.

Sounds like it is worthwhile during new root formation, which is what you're talking about in the first few weeks.

On the topic, can you make willow water equally well with both green and dormant willow twigs?

I donb't know, but it is a substance which is in the bark.