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Best I can make of it is that it began with the 1979 revision of the (Episcopalian) Book of Common Prayer, when the liturgists decided since Eucharist was replacing Morning Prayer as the standard Sunday worship service, there needed to be a more formal introductory rite. In the 1928 Prayer Book (and in earlier books), the rubrics assume that Morning Prayer and its introductory rites would have preceded Eucharist in a similar fashion as older Catholic rubrics assume that Terce would have preceded Mass. In both the 1928 and earlier Anglican Eucharist services and the Tridentine Mass, the priests do not formally greet the people, while in the preceding offices (Morning Prayer/Terce) there is a formal greeting ("O Lord open Thou our lips"/"Deus in audiutorium"). In order to make the service make sense when standing alone, the liturgists compiled 3 different greetings for the 1979 Prayer Book, and I do not know why the specific greetings were included because they are not found in any tradition related to the Anglican liturgy or any other Western rite that I am aware of. Perhaps there is an explanation for it in Hatchett's exhaustive Commentary on the American Prayer Book, but I do not have a copy on hand right now.
The practice of using the Paschal greeting became more and more common over the years within the Anglican Communion and in other Protestant Churches as well, and even began to be used outside of the liturgy.