I think it depends on the school. Most of the schools set up a different "deadlines" section just for Consortium like Stern (
https://www.stern.nyu.edu/programs-admis ... /deadlines). Stern's is the same timeline as regular applicants.
On the other hand, I know UCLA just posts a general deadline (doesn't distinguish between applicants) and I talked with an admissions officer that said that Consortium applicants find out later than regular applicants. However, that was when the Consortium app was due two weeks after the regular apps. This year it's before so who knows!