@Adiii
What doorbell do you actually have?
The "Nest Doorbell (battery)" (https://store.google.com/gb/product/nest_doorbell_battery) can only be installed in the Google Home app, but your post says "Nest app (iOS)". If this doorbell is wired to your doorbell wires, then its battery should always be trickle-charging (but can have problems charging in cold weather: https://support.google.com/googlenest/answer/11830989), but the doorbell also comes with a short USB charging cable that you can plug into a USB charger; it plugs into a small USB-C port on the back of the doorbell. If your doorbell transformer doesn't meet specs (https://support.google.com/googlenest/answer/9247132, it might not be sufficiently charging your doorbell's battery.
The 1st gen Google Nest Hello Doorbell (we have 3 of them) gets installed in the Google Nest app, and does have a small internal battery that is only used to power the doorbell during the instant when the doorbell button is pressed and power needs to be diverted to ring your internal doorbell chime. The battery on all 3 of our Nest Hellos has failed, causing the doorbell to go offline for a minute whenever the doorbell button is pressed, so we've turned off the "Indoor chime" option (preventing the doorbell from going offline) and use "Visitor announcements" instead. But to my knowledge, the Nest Hello Doorbell never sends out a battery dead message, and its internal battery cannot be user-charged.
Neither doorbell is designed to be opened up for user maintenance.
