New ignition won't turn over

Is that a new cylinder or did you put the old one into the new switch assembly? If it's an old cylinder being reused, why were parts changed? What was the previous problem? A common cause of everything else working except for a failure to crank is a cracked cam on the end of the cylinder. When it's loose, it won't turn the switch quite far enough to hit the starter contact. The dealer's parts department has a simple repair kit for that.

For other symptoms, most tests can be done at the starter relay socket with a test light or voltmeter.

Lets test before we replace. Look for a 14 gauge yellow wire in the switch connector. It should have 12 volts when the switch is in the "crank" position. If it does not, check for voltage on the 10 gauge red wire. If it's missing there, suspect a blown fuse, # 10, a 50 amp under the hood.

Fuse 10, a 50 amp under the hood feeds that part of the ignition switch, but it also feeds the wipers and radio in the run and accessory positions. If those things work, the fuse is good.