not going to be a fuse.
If you think about it, if a fuse where blown that would be it, the thing wouldn't work till you replaced the fuse.
As its coming on after a period, in electronic terms the suggestion is something is eventually heating up enough to start working, maybe a dry joint or a faulty component in the radio.
In my old sony radio it was the power transistors in the internal amp.
There are places you can send them for repair, but it may be just as cheap and easy to buy another OEM radio off ebay or fit something a bit more up to date?