Is the overdrive control pushed in or out? If it's in, try pulling it out (with car not moving of course) and see if it will go into reverse. I had the same thing happen. I could actually pull the lever into reverse but I had to hold it there. If I let got, it would bounce right back to neutral. I could use reverse, I just had to hold the lever in that position while backing up. If I pushed the overdrive cable in, the car would then stay in reverse when I shifted it there. There is a lock out switch on the transmission which breaks the overdrive circuit to prevent the overdrive from engaging when the car is in reverse. When you put the car in reverse, there is a rod that slides back and pushes this button in to break the circuit. If anything impedes that rod from moving all the way back, it will not allow the transmission to fully engage the reverse gear.
I've heard others claim this was caused by the return spring for that rod being broken. I also remember a '51 Frazer where the owner installed a backup light and mounted the switch down on the transmission. The switch moved forward slightly over time and eventually prevented that rod from moving all the way back- same symptom, car would not stay in reverse.
Unfortunately for me, when this symptom occurred on my Henry J, it was due to a piece of the casting on the overdrive adapter plate breaking off allowing the large snap ring to come loose and that blocked the rod from going all the way back. This required tearing the transmission apart and replacing that adapter plate which I was very fortunate to find.