The three cops at the center of the NYPD milkshake “poisoning” scandal never even got sick, and there wasn’t the slightest whiff of criminality from the get-go — but that didn’t stop gung-ho brass from rolling out the crime scene tape and unions from dishing out empty conspiracy theories, The Post has learned.

The fullest picture yet to emerge of the incident — which came amid fraught tensions between the police and public — is based on records and multiple interviews with police sources.

The three officers were down from The Bronx, on protest duty in Lower Manhattan, on June 15 when they ordered the now-infamous mid-shift treats via a mobile app around 7:30 p.m., purchasing three shakes across two separate orders, sources said.

Their drinks were waiting for them when they arrived at the Shake Shack on the second floor of 200 Broadway a few minutes later. A female officer picked up her strawberry shake from the front counter, and her two male colleagues grabbed their cherry and vanilla drinks from a side counter, sources said.

Police sources explained it was clear that the workers couldn’t have known cops had placed the orders “since it wasn’t done in person” — and they couldn’t have dosed the drinks after the officers arrived, because they were packaged and waiting for pickup when the trio walked in.

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