Are you sure it was the regulator ?
A Squareish looking lump of alloy with fins on one side and 3 wires on the other .
Did the engine start or did the unit cook while you were cranking the engine ?
I gather this is a used mower.
The starter should have a heavy red wire connecting it to the starter solenoid then a similar red wire connecting the solenoid to the + side of the battery.
The regulator should have a thinner red wire going to the battery side of the starter solenoid.
