Hi scenn,
Thanks for reaching out and sorry for the delay.
Our Nest thermostat's internal battery won't charge if it's not receiving power from your heating, ventilation, and air conditioning (HVAC) system. In most cases, it will show a code or message on it's display which will help you determine the cause of the concern. Please check on this link for more information as well as some suggested troubleshooting steps that you can try: https://bit.ly/3mxcwp6
To answer your other question, the green light will still blink even when the battery is fully charged. Usually, your thermostat will take about half an hour to recharge. But if the battery is fully drained, it can take as long as 2 hours to recharge.
Best,
Melba
