It depends on many factors, such as experience, learning abilities, and how well people work together. Speaking for myself, I can start becoming useful within my first morning using a new language or technology. Then, given that I'm working with people who enjoy sharing their knowledge, I can be up to speed with them in a week.

What I tend to dislike with the concept of prior knowledge before being let to work on something, is that in the vast majority of cases, once you join a team that has such a requirement, you notice their knowledge acquisition rate is currently flat; they've become stagnant themselves and are generally insecure about what they know and don't know; it's usually people with a currently flat learning curve who assume others need to know a technology prior to working on it; they're just projecting their static mindset onto others.

Some people eat new programming languages for breakfast if they have to. Deal with it?