My car’s radiator is full of oil, no water in oil, and no smoke whatsoever from the exhaust. Tell me the bad news?
Any chance someone put oil in by mistake? Beyond that, it takes hands on inspection to make sure.
Check the automatic transmission fluid. The transmission cooler is incorporated into the radiator, and leaks between the two can develop.
It’s proboaly transmission fluid. If so, new radiator needed. Good coolant flush. Transmission fluid has been contaminated. Got to get all the transmission fluid out and flushed. If your lucky, you haven’t damaged the engine or the transmission. I wouldn’t drive it like this for the sake of the transmission.
if it’s really oil, you have a blown head gasket. Remember to make sure they mill the head when they put the new one in so that a proper seal can be made. some shops skip this step.
If it’s a small amount of oil, it may be the head gasket, but if it’s a major amount, then the head itself may be cracked. Have a reliable mechanic check the gasket and head.
You aren’t using any BP products, are you? Do you see any shrimp or oysters in there? 
Has the govenor of Louisana stopped by?