Typically this will only last a minute or two. If it takes much longer time check to make sure the display is correctly connected to the thermostat base.

Nest Cam Tips And Tricks Get The Most Out Of Your Nest Cameras
Nest light blinking green. Battery is too low to start. The nest cam is working someone is watching the. The nest cam has been plugged in to a power outlet and is booting up or re booting. If the lights continue to blink for more than 5 minutes remove and reconnect the display. A small flashing green light on the nest thermostat display usually means that its updating software. If your nest camera is blinking green then you should know that your nest cam is working properly and is connected to the internet and you can watch the live video feed.
When this happens your nest thermostat cant draw power from your system to charge its battery. Blinking green light wont go away any ideas on this one. A blinking green light near the top of your nest thermostats display means that its updating the software starting up or restarting. Your thermostat has a green blinking light if a green blinking light suddenly appears it just means that your nest is automatically updating its software. During this process the display will restart which usually only takes 1 2 minutes. Nest camera blinking blue.
A blinking green light will appear if the nest is turning on restarting or updating. You can turn off this light via the control screen. It should automatically turn on when finished. It should automatically turn on when finished. Green light has always been used to indicate that everything is good and the same is with the nest camera blinking green. If the light is still blinking after reconnecting the device restart it by holding down the thermostat ring for 10 seconds.
The nest cam is working and someone is watching its video stream. You may also see a furnace heads up message on the thermostat screen in your nest app messages or. Got a replacement nest thermostat 2nd gen from ebay. The thermostat shows a notification that says please remove the thermostat from its base then reattach it the thermostats display wont light up and unresponsive you may also observe a blinking red or green light above the display the thermostat cant control the corresponding heating and cooling units. But if this continues for longer than a few minutes see the article below for more troubleshooting. Its replacing an existing 2nd gen so i think im safe to assume the wiring is correct on the base.