In "Delenda Est"

Three Successive Timelines
(i) The Romans win the Second Punic War.
(ii) Neldorian intervention causes Carthaginian victory.
(iii) Patrol counter-intervention restores Roman victory.

Manse and van Sarawak travel futureward from the Pleistocene lodge, expecting to arrive in New York, 1960 (i). Instead, they arrive in (ii). After traveling pastwards and counter-intervening, they are able to travel futureward into (iii). It follows that they never arrived in (i).

They had intended to complete their furlough in 1960 (i). After that, Manse would have remained in New York whereas van Sarawak would have returned to 24th century Venus. In fact, neither arrived. They had gone to the lodge and never returned. Someone from their home offices would have traveled to the lodge to investigate and thus would have become involved in the attempt to restore Roman victory.

"(They never told you when you were scheduled to die, and you had better sense than to try finding out for yourself. It wouldn't have been certain anyhow, time being mutable. One perquisite of an agent's office was the Danellian longevity treatment.)"
-Poul Anderson, "Delenda Est," section 1, IN Time Patrol (New York, 2006), p. 175.

(An aside: Here we learn that the longevity treatment comes from the Danellians, not from future human medical advances.)

Manse thinks this before leaving the lodge expecting to arrive in New York, 1960 (i). Thus, when he thinks it, he is in a timeline where there is a recorded date not for his death but for his disappearance while on vacation at the lodge.