Waiting to hear back from a professor—how long should I wait?
This is business, send a nice and polite email to your professor to ask him if he has news since you don't.

Don't wait 3 weeks on a job application. 

Obviously, maybe he'll say 'it's summer everyone is on vacation', maybe he forgot, maybe his colleague forgot. Show that you're willing to work with them, just don't spam their mailbox every couple days.