So my question again: could they begin demanding “discovery” [of specific targets of inquiry*] TOMORROW?

Not tomorrow. But certainly soon. It's been a number of years since I had to deal with basic Civ Pro. I know that you don't have to wait for all the motions on the pleadings to be done before you start discovery, but it doesn't start right away. There's a bunch of preliminaries at the start of the lawsuit - service of process, getting the court docket set up, and at least one early "meet and confer" session between the parties to just sort of lay out how things will go. But that wouldn't take more than a month or two.

But the Times wouldn't gain anything by "demanding" discovery regarding these things. Trump wouldn't turn anything over. His team would file motions arguing that they don't have to disclose certain things (not relevant, protected by privilege, etc.). And the NYT would file motions saying, "yes you do." And it would take several months for those motions to get resolved. Because, again, the judge is not going to be in any hurry to resolve these discovery disputes in a case that will probably be dismissed under basic First Amendment grounds.

There wouldn't be any counter headlines. "Judge delays ruling on routine motion for protective order in discovery" isn't going to make it into any newspaper.