I understand that the scooter only turns on when the battery charger is connected to it, and when the battery charger is unplugged from the scooter that the scooter does not turn on.
This could be caused by either a problem with the connectors, fuse, or circuit breaker between the battery pack and controller, or from a faulty battery in the pack. Most of the time when this happens with new batteries it is from a loose connection, blown fuse, or faulty circuit breaker.
The first step we recommend to take is to inspect the connectors which attach to the batteries, and also the connector on the other side of the battery pack wiring harness which attaches to the speed controller. These connectors should not be discolored or melted, and should be firmly attached the the battery terminals or speed controller connector. If the connectors are loose on the battery terminals they can be removed from the terminal and closed up a little with pliers and then reinserted back onto the terminal for a better fit. A definitive test on the battery pack wiring harness and fuse or circuit breaker would be to unplug the battery pack wiring harness from the controller and test to see if it has Voltage at the connector that was unplugged from the controller.
If all of the battery pack wiring harness terminals and connectors have passed inspection and been adjusted to fit tightly then the next step to take is to test or replace the fuse, or to test or replace the circuit breaker, depending upon which type of circuit protection the scooter has. Fuses and circuit breakers can be tested for continuity with a multimeter to see if they work or not, or replaced with new parts if you don't have or want to purchase and use a multimeter. Here is a video we made showing how to test fuses. Testing circuit breakers is the exact same process as testing fuses so this video can be used as a guide to test either of these parts.
If the battery pack wiring harness and its connectors test good, and the fuse or circuit breaker test good, then the next step to take would be to load test the batteries to see if one of them is faulty and causing the problem. Here is a link to our electric scooter battery load testing guide: http://www.electricscooterparts.com/loadtest
Performing these tests and procedures should narrow down the part that has failed and is causing the problem, however if the problem still persists then please let us know and we can then offer further suggestions for troubleshooting and repair.
Please let us know how it goes.