SM Stirling exactly reproduces Manse Everard's way of thinking about variable realty. Trapped and imprisoned in the 1926 of a divergent history, Manse thinks:
"Either a Patrol rescue team would arrive to break him out...or he'd vanish when this world was cancelled." (Multiverse, p. 77)
(For convenience, I am referring to the Patrol timeline and the divergent timeline as timelines 1 and 2, respectively.)
"...when..." does not refer to any temporal coordinate within timeline 2. Timeline 2 either does or does not exist. Since Manse is in it, it does exist. It is a continuum with one temporal and three spatial dimensions. Its spatial dimensions extend to the edge of the universe or around the curve of space or etc. There is no reason to suppose that its temporal dimension does not extend pastward to a Big Bang and futureward to a heat death of the universe. If this timeline ends prematurely before the heat death, then its premature ending is a random event not caused by the activities of any time travelers.
If a time traveler prevents the prevention of the Sarajevo assassination, then the consequence will not be that timeline 2 exists until sometime in or after 1926, then ceases to exist. The consequence will instead be the creation in a second temporal dimension of another four dimensional continuum, this one containing the history guarded by the Patrol. It will be timeline 3, differing in only a few unimportant details from timeline 1.
From the point of view of the inhabitants of timeline 2, timeline 3 exists/will exist (a Temporal language tense is needed) in the future not of timeline 2 but of a second temporal dimension at right angles to timeline 2. Each moment of the second temporal dimension contains a four dimensional continuum just as each moment of the first temporal dimension contains a three dimensional universe.
The continued existence of timeline 2 after its version of 1926 is confirmed on p. 71 when Wanda travels two centuries, then another two centuries, futureward. If Manse is not rescued, then he will not "vanish" but will live the rest of his life in timeline 2. Of course, people in timeline 3 will have no access to Manse and might consider that he has vanished but there will have been no vanishing from his point of view.