Ouch. The oil/water emulsion may form a gooey mess that will be difficult to flush out. Your water pump may be trying to pump something that looks like whipped cream. The good news is that it is not uncommon for transmission fluid to get into radiators. It happens when integrated transmission coolers start to leak, so there may be tips on the internet for flushing it out.
After checking Google for tips, I would first try flushing it out with lots of distilled water, and perhaps some commercial radiator flush. Those should not damage the system and won't make it worse. Good luck.
If that does not work, I would suggest mixing some coolant/water/hydraulic fluid in a jar and see what forms. Then try different solvents (e.g. different kinds of alcohol) to see what will dissolve the gunk.
The concern here is that there are various plastic parts, hoses, O-rings, and gaskets in your cooling system that may have chemical compatibility issues with anything you put in there, so you don't want to pour in anything that does not help and might harm. Your mechanic and maybe even the manufacturer probably won't be able to tell you with confidence what solvents might help or hurt.