This is bringing back memories. Sorry to hear about your problem. I recently had this happen to me and learned a lot from this forum's help. I think I was fortunate to save the transmission and coolant did not get back into the transmission...plenty of transmission fluid in the radiator though. I captured a sample of the transmission fluid from the transmission cooler lines and coolant from the radiator drain plug, put samples in glass jars and watched them separate over time.
White lid is coolant sample, black lid is transmission sample. Took sample to transmission specialist in my area and he said even the slightest amount of coolant in the transmission fluid would change the fluid color and ruin the transmission in short time. He said I was good to go.
Did you lose the transmission, or do you
think you lost the transmission because there was transmission fluid in the coolant?
With the thermostat removed, I flushed out as much of the old contaminated coolant as possible from the upper hose and out the bottom hose into a bucket. Installed the new radiator and ran a cycle of distilled water through system. Drained, and removed all remaining fluid by venturi cooling system vacuum. Refilled with distilled water and Prestone radiator flush and degreaser. Followed procedure.
Radiator flush and oil degreaser
Another couple drains and refills and venturi vacuum of system with distilled water. Cooling system is looking pretty clean now.
When I am done with other work that has arisen from this problem, I will refill with 50/50 coolant.
Other problems - Had to replace hoses and seals because the transmission fluid compromised them. One leaking seal in particular was feeding a heater core tube under the intake manifold, so I needed to remove intake and water pump to get to it. It was leaking fluid on top of my block under intake. Replacing water pump (seal originally, but decided to do pump since it was cheap and already off), replace t-stat and t-stat gasket, lower radiator hose seal came with new radiator.
Then drain torque converter, drain transmission fluid, remove and replace filter, refill transmission system.
Good luck and pay attention to those hoses and seals...they are relatively inexpensive, so probably a good idea to change them anyways and watch out for extra holes in the ends of the hoses where they pierce for hanging straps, find a source where they use clips to hang instead of poking holes in hoses.