The Nest doorbell connects via 2.4 GHz or 5 GHz WiFi to Nancy's router. Also connected to the router is Nancy's phone. The doorbell communicates through the router to the phone, and the Nest App receives the data packets (video, audio, alerts) and displays the info.
While the Nest doorbell info online says the A/V feed goes directly from the doorbell to the phone, it clearly also goes from the doorbell to the router and through the internet to Google's servers. And that's true with or without a subscription, at least to some extent.
When LE mentioned the Nest doorbell disconnecting at 01:47 and a person appearing at 02:12, they hadn't yet recovered any video footage. So, when they said the doorbell disconnected, they probably meant it was unable to communicate with the phone for some reason, likely because of WiFi connection failure or jamming. I don't think they meant that's when the suspect removed the doorbell from the porch because at that time, LE had no way of knowing for sure what time that happened.
LE might've known the disconnect/human seen times even before having the phone forensically examined because the alerts would've been showing on Nancy's screen. When they learned more later (e.g., when the doorbell reconnected), they decided to hold back that info. IMO