I have soldered up a solution to effectively bypass the Nest API - or lack of for the actual push button.
Instead of connecting the Chime Connector to a hard wired old door bell chime iIhave a 16v transformer power source and the Chime Connector connected to a simple non-latching relay.
I them got a Z-wave Water sensor, cut off the probe and use the open wires to connect to the relay: Normally closed terminals.
Change your device type in the Smartthing api to a z wave button and your done. Now when the Door bell is pressed the Relay activates the Z wave module and you get a notification on your smartthings.
I then set up a Echo Speaks Webcore Piston to find a Door bell chime on Apple Music that rings triggers when the button is pressed.
I also used the two normally open 2 terminals on connect a hacked push button for a standard wireless doorbell & ringer.
My transformer, z wave device and Wireless Doorbell button all go into a double din blank socket box located beside a standard wall socket box and powered from there.
The Door bell wires them simply pulled through the wall as normal.