In an empire as vast as that of the Romans, how long would it have taken for the news of an Emperor's death to spread to every province? Would some provinces not know of the Emperor's successor and question their authority under the false belief the old Emperor was still alive?

Stanford has a really cool tool called Orbis which can calculate the distance and travel time between points in the ancient Mediterranean. There are various variables you can control such as type of conveyance, season, route type, et cetera. "Under the hood" is a lot of very good background data establishing a dataset of ancient travel. In this scenario, we are talking about the death of an Emperor, the news of which would travel out to the provincial administrations via Imperial rescript, a very rapid courier system similar to the early American Pony Express. On Orbis, this represents a land travel value (depending on the extant roads) of about 250km per day. When faster, routes via fast ship are calculated instead or in addition. We can then ask Orbis to make some basic calculations for the absolute fastest a message could be carried from Rome to various points. Using the tool and going generally counter-clockwise around the Empire from Cadiz:

-The Fastest journey from Roma to Gades (Cadiz) in July takes 10.3 days, covering 2449 kilometers. — The Fastest journey from Roma to Gades in January takes 10.1 days, covering 2449 kilometers.

-The Fastest journey from Roma to Carthago in July takes 4 days, covering 627 kilometers. — The Fastest journey from Roma to Carthago in January takes 3.5 days, covering 625 kilometers.

-The Fastest journey from Roma to Alexandria in July takes 11.4 days, covering 2418 kilometers. — The Fastest journey from Roma to Alexandria in January takes 10.8 days, covering 2412 kilometers.

-The Fastest journey from Roma to Antiochia (Syria) in July takes 12.4 days, covering 2672 kilometers. (Surprisingly, seasons don't seem to impact the time very much, so I'll stop checking them).

-The Fastest journey from Roma to Circesium (a Roman fort on the Euphrates frontier) in July takes 14.8 days, covering 3284 kilometers.

-The Fastest journey from Roma to Aquincum (Budapest, on the Danube limes) in July takes 5.1 days, covering 1152 kilometers.

-The Fastest journey from Roma to Augusta Treverorum (NE Gaul) in July takes 7.5 days, covering 1381 kilometers.

-The Fastest journey from Roma to Londinium (Britain) in July takes 9.2 days, covering 2081 kilometers.

And so, not surprisingly, the longest lag time would be to the far eastern frontiers, at about 14 or 15 days, perhaps 20 days or so at the maximum. The time elapsed between the death of the emperor and a new successor would be no more than the lag between those events in Rome, given that subsequent rescripts would be despatched with the corresponding news. So in essence, nothing would happen differently in the provinces compared to Rome as far as any confusion in the transfer of power. If a local Roman official decided to dispute the outcome of a succession (like Vespasian, e.g.), then their decision would happen in "real time" more or less, but on a delay determined by the amount of time needed to travel back and forth.