According to the OED
1947 Sci. News Let. 2 Aug. 70/1 Emergency spurts of speed of jet-propelled combat planes will result from a development of the Ryan Aeronautical Company which the makers call an ‘after burner’
The same source refers to 'reheat' as
[1947 Pop. Sci. Monthly Oct. 87/1 As a next step, they've devised the after~burner. It's principle, essentially the same as the steam-turbine reheat system, is fairly simple.]
1948 Brit. Patent 606,176 4/1 The act of turning on the re-heat will open the propelling nozzle.
1949 Flight 8 Sept. 285/1 Exhaust reheat, or ‘afterburning’, is the name given to the process of burning fuel in the exhaust pipe of a jet-propulsion unit.
1972 D. Hart-Davis Spider in Morning ii. 21 The reheats were in and burning fuel at a terrifying rate.
2005 R. R. Lawrence Mammoth Bk. Space Explor. & Disasters i. 4 The XF.91 was powered by a General Electric J.47 turbo-jet engine equipped with reheat
So it looks like 'afterburner' was first with 'reheat' coming later