I think the method here is for Trump to give Miller 45 or 60 days to present his evidence in final form and on the 46th or 61st day if there’s nothing there the whole thing goes away. End of story.
I don't know how much of this will be public to start with, but I don't foresee a deadline being imposed from the WH.
Lots of ways this could lay. Sessions could write Rosenstein and demand assurance that Mueller is NOT working on any investigation other than "the campaign," other than whatever Sessions has recused from.
Sekulow, Cobb and Dowd can do the same thing, to Rosenstein and to Mueller.
Congress has oversight over the DOJ, and should be upset if a Special Counsel usurped the role of the DOJ, or stripped sessions of a power that Sessions had not relinquished. I doubt Congress does anything though, Congress all but endorsed Comey usurping the role of prosecutor.
Sekulow Cobb and Dowd can also take to the press with their arguments. OpEds, appearances on news programs. Ask the people to complain to Rosenstein, flood his inbox, swamp him, bury him with attention. Turn on the spotlight.
I don’t think there is any deadline as part of Mueller’s charter. Barring actually finding anything they want to keep dripping leaks to keep to the heat on until 2020. Sessions f’d us but good so there is no good way to stop this now, the best they can do is shut it down with a storm or a total President killing S-Storm.