That is a nice goal. Did you know that Adafruit does a lot with that ? Including many Youtube videos. The Wokwi simulator can do CircuitPython on a RP2040. It is still in development, but it is fun.
So they did actually made it pin-compatible. That is very nice, I didn't know that.
I have been looking into that Servo Shield before. There was something with it, I'm not sure (I think it uses the voltage regulator of the Nano board to make 5V and then it has an extra 3.3V voltage regulator to bypass the 3.3V on the Nano board).
VIN can be 12V and above for both the classic Nano and the Nano RP2040 Connect.
The digital output pins are 5V 20mA (40mA shortcut current) for the original Nano and 3.3V 4mA for the RP2040.
As far as I can tell, the only problem seems to be the 3.3V level of the signals. You need Servo Motors that can use that signal or use hardware to amplify the signals. Some Servo Motor explicitly tell that they are 3.3V signal compatible: https://www.pololu.com/product/3421.
Do you have a schematic for the servo board ? I hope your RP2040 is not fried.