An 18 V battery mower will almost always cut off at some voltage to protect the battery, and I would expect that voltage to be somewhere around 14 to 16 V.
A 12 V battery charger will often put out 18 V with no load. Chargers designed to connect directly to a battery are unlikely to be suitable to run anything directly.
The current that the mower will take, especially when starting, will be quite big and you would need a very large power supply to run it.
Connecting a power supply backwards may have damaged something but without a lot more information it's hard to know for sure.