The 1250 has a couple of safety interlocks on it, that if they are not working or are not correctly triggered, the key won't do anything just like you describe. First, before doing anything, make sure you've got good ground to the engine, and make sure the battery is good, fully charged, and load tested. I had one this spring that showed 12.6 v with the key off. Turn the key to ON, and it dropped to 3 v. Turn the key to START, and it dropped to 0.5 v. Turn the key off and it's back to 12 v. Took it in to NAPA and they couldnt even load test it, dead short in the battery! So, I'd double check it before doing anything.
Safety Interlocks.
1. Clutch pedal must be depressed while trying to start. There's a button switch on the frame cross member under the tractor, and a piece of spring steel on the clutch pedal rockshaft, so that when you depress the pedal, the spring steel presses on the button switch. This little piece of spring steel often bends or breaks off. I'd check it first. You can jumper the switch with a U shaped piece of wire, and shoving it in the socket that plugs in to the switch. This will take it out of the equasion.
2. Electric PTO switch must be off. This one is a whole lot more difficult to figure out. There's not an easy way to jumper the switch to take it out of the equasion. First try turning the key to the ON position, then toggling the switch all the way forward like you are going to engage the mower. You should hear the magnent in the PTO "clunk". IF it does not clunk, there's a pretty good chance the switch is bad.
If you look online you can easily find a wiring diagram if you need one, but this will get you pretty far!
Bill