Assuming Nightmare was chosen for Conspiracy, all "[c]reatures you control"—
are Nightmares (and no other creature types [C.R. 205.1a]), and
get +1/+1,
regardless of the order in which Chainer and Conspiracy entered the battlefield. This is because the two effects apply in different layers (layer 4 for Conspiracy, and layer 7 for Chainer's first ability), so that dependency and timestamp order are irrelevant here (C.R. 613.7, 613.8, 613.1d, 613.1g, 613.4c).
EDIT (Mar. 23, 2021): Edited rule citations, including because some rules were renumbered in the meantime.
What is your question?
There are two possibilities:
1. If Conspiracy enters after Chainer's ability resolves, the creature will be the Conspiracy chosen type and not any other types.
2. If Chainer's ability resolves after Conspiracy enters, the creature will be both the Conspiracy chosen type and Nightmare type.
EDIT (Mar. 23, 2021): Edited rule citations, including because some rules were renumbered in the meantime.
Chainer's timestamp is irrelevant.
The relevant timestamp is the moment its ability resolved.
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