There is a fuse for the blinkers, located under the dash, between the key switch and the flasher. Attached is a photo of a 54 hardtop with the dash removed. Also attached is a snapshot of the wiring harness from the blue shop manual, which is a useful reference (although it's completely void of wire color..).
You should verify that all bulbs and the flasher are good, and use a lighted probe to check for shorts. You can replace a fuse, but it may well blow again if the ultimate root cause is not addressed. The old cloth wiring could be frayed and touching ground somewhere it ought not to.
Good luck