Nano-3 will not charge

This is from the Nano³ manual:

4.11 Replacing battery
The battery can be replaced by user. You will need a screwdriver with cross head (Phillips).
Unscrew three screws.
Open the cover and remove the battery.
Insert a new Nokia 3310 battery (Li-Ion). Make sure that the contacts are properly aligned.
Close the cover and screw back three screws gently.

So no problem there. It could be that your new battery is less resistant than your first one, especially when getting very low on charge...

The only problem I had with my Nano³ (original battery - 2014) was that it disconnected in flight, interrupted the log and restarted the Nano. I solved this by putting some padding behind the battery, to keep it in place..