it looks like I’m going to beat my deadline, will my publisher move up my pub date?
Probably not. It might happen if there was a compelling reason, a breaking news story, for instance, but for the most part your publisher’s resources are allocated among any number of books, and your publisher has to coordinate editorial, author review, production, and sales and marketing for every title (and staff is finite). Not to mention that the retailer expects to see you on a specific date as well.
It doesn’t hurt to be early: you get a great rep, you have plenty of time for editorial development, but you probably won’t speed up publication.