"Can Access be set up to deliver time based alerts if certain criteria are not met? "
Hmmm. Access is really designed for data storage and the manipulation & reporting of that data, however if you have someone clever enough then I'm sure they can use VBA Code in Access to get it to do this.
"I need something to not only allow shared access to the information, but to set permissions so that only people from department x can update information that pertains to dept x. I also need to drive reports on the data to communicate the progress of jobs within the timeline and set accountabilities to departments for their tasks and communicate (automatically) to them that tasks are available to them or overdue and so on. "
Now this is more like it, databases are designed to be multi-user with permission settings and to create reports so YES to all this.