This seems both interesting and hard to pinpoint. Köck ([1991](//zbmath.org/?q=an:0731.19005); [1998](//zbmath.org/?q=an:0951.14029), 4.6b) credits Fulton-Lang ([1985](//ams.org/mathscinet-getitem?mr=88h:14011)) and Soulé ([1985](//ams.org/mathscinet-getitem?mr=87b:18013), *cf.* Thm 7), but your suggestion that the theorem must have been “known” earlier is also well-supported. For one thing, Rössler ([1999](//zbmath.org/?q=an:0961.14006), [§1](http://people.maths.ox.ac.uk/rossler/mypage/publi.html)) finds it in Manin ([1969](//zbmath.org/?q=an:0199.55502|0204.21302), [Thm 16.6](http://mi.mathnet.ru/eng/umn/v24/i5/p3)). For another, Dyer ([1962](//zbmath.org/?q=an:0171.43901)) (cited by Eckmann at ICM ([1963](//zbmath.org/?q=an:0129.38802)), Adams ([1965](//ams.org/mathscinet-getitem?mr=33:6626), (iii) p. 152), Fuks ([1973](http://mi.mathnet.ru/eng/inta/v8/p71), pp. 349–350), and reprinted in Adams ([1972](//zbmath.org/?q=an:0234.55002))) starts:

> This lecture is principally an exposition of a folk theorem of
a Riemann-Roch type for general cohomology theories known to
Adams, Atiyah, Hirzebruch... .

While these authors don’t spell out how this “folk” theorem includes yours, Panin ([2004](//zbmath.org/?q=an:1066.14010), [§0.1](https://faculty.math.illinois.edu/K-theory/0552/); [2004](//zbmath.org/?q=an:1077.14501), [p. 823](//www.mfo.de/occasion/0412/www_view)) says at least that his Theorem 2.5.3 “inspired by a Riemann-Roch theorem from [[Dy](//zbmath.org/?q=an:0171.43901)]”, does. See also Smirnov ([2006](//ams.org/mathscinet-getitem?mr=2008h:14009), 2.5.3).

So my impression is that specialists understood the Adams Riemann-Roch theorem as an instance of the more general **Dyer**-Riemann-Roch theorem, long before any of them bothered to name it.