Corrosion, Rust, Dirt in the sockets, connectors, switch and blinker unit, as well the wiring itself.
Radio waves are made by running electricity thru a wire (antenna). If the current changes in a way to create a signal, the signal is transmitted thru the air as radio waves.
Something has changed in your blinker circuit that instead of current switching on and off completly, it is now switching on and off with a ripple effect and creating radio waves the radio will pick up. You can't see the unsteady current that is rippling down the wire in the light bulb, but you can hear it on the radio.
Check and clean up all the stuff in the electrical circuit, maybe a new blinker unit will fix it. As mentioned, corroded or damaged grounds can do the same thing.
How bad is the noise over the radio, this is very much one of the things you can live with.
As well, most of the 80's on OEM Radios and all the after market radios were built to be very sensitive to tune in distant stations. I'll listen to WABC (NYC) all the way down in Southern Maryland, the tuner in the radio turns its sensitivity up to max to get that distant station in and when listening to the station I will also pick up noise from my turn signals, heater/vent fan at max, power lines along the road and even micro-wave ovens at fast food places I drive by. In cases of tuning in distant stations with the tuner at max sensitivity its not the electric systems making to much radio noise, its your radio tuner set to such a high sensitivity it is picking up the slightest noise out there.