Craig has got it right. We would first check for Voltage where the battery pack plugs into the controller. There should be around 52 Volts there. If there is no Voltage then the fuse may be blown or the circuit breaker may be tripped. If the controller is getting the right amount of Voltage then the next thing we would do is unplug the power switch from the controller and check continuity of the power switch to make sure it is working.
Installing fuses on both the positive and negative battery wires would not cause any problems.
The speed controller may have labels on the plugs that can be seen on the inside connector after the connectors have been disconnected from each other.
The power or key switch wires are usually red and black.
Oh wow, I have never heard of a scooter with a side stand (kickstand) switch before. That is very good to know about.
I am not sure how to go about testing the battery cells. Most of those electric scooter lithium battery packs do not have balance leads and are in a plastic case or covered in heat shrink plastic tubing so the individual batteries cannot be accessed.