If the radio is wired to work with the ignition off, then it seems the problem is in the ignition switched circuit. An auto electrician may be able to trace it.
You need to establish if the ignition switch is getting an uninterrupted supply. If the radio feed is through there then it is.
I would check the connections in the column shroud area. Always a tight fit and prone to dodgy connections in many cars.
Then check the ignition relay. If these check out it gets harder from that point.
