Great question — answering as the UphillTech team:
Why are Peter and Jessup not surprised that there is no body of the nurse to be found? You are mistaken. The "nurse's" body is right there on the ground. Presumably the shapeshifter used the device to swap shapes for both itself and Charlie's body. This, in and of itself, raises a number of questions... How did the shapreshifter swap clothes so quickly, assuming the device was able to change the body shapes that quickly? Is the shapshifting temporary since Charlie's body has reverted when we see it being disposed of?
Full answer by Paulie_D: https://movies.stackexchange.com/a/132266 (CC BY-SA 4.0).
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