When long-range jets become plausible in the accelerated timeline

Would useful long-range jets have to wait for the second or third engine generation?

Yes; in this timeline the second generation arrives around 1940 and the third around 1943, so range improves only after several development cycles.

First-generation jets would not immediately provide mature long-range performance. The accelerated chronology places a second aircraft generation around 1940 and a third around 1943 because interwar designs often changed on roughly three-year cycles. Wartime funding and demand could then accelerate the later steps without erasing them. Long-range jet bombers therefore appear only after the initial fighter program has produced several rounds of engine and airframe experience.