This merger is thought to have happened in the 5th-4th centuries BC. More specifically, it's a monophthongization, in which [ei ou] became [e: o:]. ("Genuine/spurious diphthongs" is a bit of a misnomer -- in Classical Greek, ΕΙ ΟΥ are not diphthongs but digraphs which stand for long vowels; some of these long vowels historically developed from diphthongs.) Here are some relevant sources.
Goldstein, "Diphthongs", in Encyclopedia of Ancient Greek Language and Linguistics:
Lejeune, Phonétique historique du grec, on ΕΙ (he later says practically the same about ΟΥ):
Allen, Vox Graeca (again on ΕΙ but with an almost identical discussion of ΟΥ later):
Similar datings are given in Buck, The Greek Dialects; Sturtevant, Pronunciation of Greek and Latin; and Sihler, New Comparative Grammar of Greek and Latin. (Sihler in one place suggests that the monophthongization of ΟΥ was earlier than that of ΕΙ, but he doesn't give evidence or details for this.)
Strangely, Horrocks, Greek: A History of the Language and Its Speakers (p. 161) dates the development much earlier:
Unless I'm misunderstanding his discussion, this seems impossible given the consistent spellings of these sounds as digraphs before the fifth century.
(A couple of relevant sources I can't currently access are Threatte, The Grammar of Attic Inscriptions, and Teodorsson, The Phonemic System of the Attic Dialect 400–340 B.C. If anyone has access to those and is interested in this question, they could be worth checking.)