Perhaps it was recorded and saved in the cloud account. Seems unlikely.
The watch supposedly has three ways to communicate:
- Bluetooth to the phone. But that's 30-50 feet max. So, too far away, if the girlfriend was waiting out on the street.
- Via WiFi, but not on just any network. Unlikely it had got on the consulate's network.
- Via LTE (cellular). But that requires support from a carrier, and supposedly there are none in Turkey.
It's much more likely the audio came from a Turkish bug, but the Turks don't want to admit to espionage (sources and methods, diplomatic fallout), so they invented the smart watch / smart phone / cloud story as a cover.
Here's one tech blog's assessment:
https://mashable.com/article/jamal-khashoggi-saudi-arabia-turkey-apple-watch/