Both of those are correct, and both are common, but they have two somewhat different meanings. You already know that one is a verb (a week to process) and one is a noun (the process takes). "Around here" doesn't really seem to fit, and I don't think you need to add anything to those two sentences. They are complete and correct.

You could say 'around one week' instead of 'about one week'. I would say though that 'about' is better if the implication is that the person knows roughly how long it will take, whereas 'around one week' to my mind implies a higher degree of uncertainty.