Surprisingly, this, taken together, is a recent innovation. "Verbum Domini" considered as the acclamation/versicle component of an acclamation/versicle-and-response to the reading of Scripture, dates from the 1970 (1969) Missale Romanum of Paul VI. (Or, since I'm not seeing it either here or here, should I say only the third typical (i.e. 2002) edition of that?)
"Deo gratias", "an ancient Roman acclamation of approval" "used not only after readings but also after the dismissal and even, in some cases, after announcements" (Marion J. Hatchett, Commentary on the American prayer book (New York: The Seabury Press, 1981), 327), was "in use by the eighth century" and assigned to "the Server" in the Missale Romanum of 1962 (and Low—but not High?—Mass before that), but "Verbum Dei" (> "Verbum Domini"?) was "suggested by Pope Paul VI" (Paul Bradshaw, Gordon Giles, and Simon Kershaw, in A companion to Common worship, ed. Paul Bradshaw, vol. 1 =Alcuin Club Collections 78 (London: SPCK, 2001), 115).
See also Paul V. Marshall, Prayer book parallels: the public services of the Church arranged for comparative study, Anglican liturgy in America 1 (New York: The Church Hymnal Corporation, 1989), 326-327, for its absence from the Anglican rite in America before 1979. Presumably it is therefore also absent at some point not long before The United Methodist book of worship of 1992 (see e.g. p. 23), the Presbyterian Book of common worship of 1993 (see e.g. p. 61), and so forth (though I haven't checked).
Cf. pp. 510-511 of the third typical (2002) edition of the Missale Romanum:
10. Deinde lector ad ambonem pergit, et legit primam lectionem, quam omnes sedentes auscultant. Ad finem lectionis significandam, lector acclamat: Verbum Dómini. Omnes respondent: Deo grátias.
11. Psalmista, seu cantor, psalmum cantat vel dicit, populo responsum proferente.
12. Postea, si habenda sit secunda lectio, lector eam ex ambone legit, ut supra. Ad finem lectionis significandam, lector acclamat: Verbum Dómini. Omnes respondent: Deo grátias.Though the revised translation of the Mass imposed from Advent of 2011 is often closer to the original Latin than the one it replaced, this (along with the inconsistent retention of "Cup" instead of "Chalice" at the anamnetic acclamation of the people "Quotiescumque manducamus panem hunc et calicem bibimus" (cf. 1 Cor 11:26), which appears in English as "When we eat this Bread drink this Cup")) is one of those points at which it is not close. For "Verbum Domini" is translated "The Word of the Lord" after the first two readings, but "The Gospel of the Lord" after the Gospel. Yet the Latin supports only the former.
My thanks to Seattle Pacific Seminary student Cory Baker for posing the question.