If the new radiator is defective, not cooling the transmission fluid down could that cause this problem? The upper radiator hose can be squeezed all the way to the middle at running temp, the lower radiator hose looks huge. My transmission got up to 238° at idle quickly today. I do have some forward motion now in 1st-Drive, no reverse feels like it's in park and neutral makes a whining noise at transmission and acts like it's in drive with forward motion. I drove a mile and a half total when pulling in my driveway that's when I saw smoke from under the hood.
this doesn't sound right. yeah, if there's no water flow across the radiator from a blockage, or somehow how the internal transmission cooler is blocked, it might explain why it's getting hot so fast and you don't have any tranny line pressure to engage the clutch packs for the gears.
that said, I've never see it happen. but anything is possibly. I know coolent is expensive but might be worth checking into.