Plot hole: SPOILER: When Amy comes back, doctors would have checked her non-existent head wound from the "abduction", from where she supposedly lost that much blood. That would and should have revealed all of Amy's lies.
Plot hole: Amy calls the Police about the shed before going to Desi. They watch Nick's TV statement together. The shed is only searched by the police after the statement. Which would imply the police sat on the report for several days before investigating.