Oh Blimey (when was the last time you heard that? lolo). Steve sir is right. There's probably coolant in your auto transmission. You cannot drive this car at all without flushing the fluid first, which is also pointless to do without fixing the main problem first.
I agree with steve sir...the transmission cooler is likely embedded into the radiator itself. The membranes separating the tranny oil and coolant have ruptured. Time to change the radiator.
When you do so, please get new upper and lower hoses, a new wp, thermostat, belts, rad cap and brass bleed screw. This is a big job but its very much in the category of long term wear and tear, so you're only fixing up your car for the next 100-200k anyway. So it should be less painful to swallow and perhaps the transmission only requires a flush and a new filter and new tranny pan gasket - which is always nice to do for your car.
I suggest that you do a fast drain interval after the flush anyway to remove as much coolant as possible from your system - use Dex 4 for the flush and the first fillup, (soak your existing tranny filter in a diesel bath for 2 hours after washing it out as best as you can with water, then reuse the filter after drip drying and soaking it for 5 minutes in tranny oil). 30 days later, drain all the oil and use Dex 6 or synthetic oil in your transmission.
Synthetic is always recommended bro but its damn pricey. However, you can do an extended drain interval with synthetic oil as it has more detergents in it and it is more resistant to shearing and general degradation. But please change the oil from time to time - synthetic oil/lifetime fluid is poorly understood and its foolish to risk a perfectly good transmission in order to save a few bucks.