Faster than you think. At work we still operate under Obama rules because a. there's few administrators b. those who are are leftover Obama appointees and c. Tillerson is nothing more than a figurehead. Nobody sees him, he doesn't go to the lower floors, and has one floor of his own he won't leave. Nobody knows who's staffing it.
Trump has fired everyone he can and can't find anyone to replace him because nobody will work for him. In the case with Bannon, it's well known here that his Russian ties are deep and he's more of a security threat than anyone.
Don't be surprised if he's the source of Russiagate. When he's forced to speak before the special prosecutor, Trump and him as well as a lot others will be facing prison.