OK, I am going to go throw a weird one at you. Maybe your problem is related to mine, but probably not. your problem is more likely a connection problem. However, I had a similar radio cutting out problem with my retrosound, except mine would cut out when I revved the engine from low speed to high, but it only seemed to happen when the car was fresh out of the garage. Also, if the car was brought back to dead idle, radio would play again -even fresh out of the garage. weird thing was, drive it a few miles, and this cut-out would go away.
So, after me living with this for a long while, and pondering it, but not coming up with anything, a friend rides with me and sees this weird behavior and has a genius insight - he says "do you have an original voltage regulator?" yes. "well then, I think maybe your radio is seeing unacceptable voltages when the alternator is kicked in at higher RPMS. The alternator kicks in much more often immediately after the car has started after it's been sitting in garage. After you drive the car a bit, the battery gets charged up - so alternator not kicking in - and symptom stops. Maybe you should try a new solid-state VR."
Freaking genius insight. Sure enough, a new solid state VR solved my radio cutting out problem completely.
Now, in your case, the radio cuts out at low speed - when another electrical system is engaged. So perhaps the current draw of the blinker system causes the voltage the radio is getting to somehow change to a temporarily unacceptable amount. Car equilibrates voltages are a few moments and radio comes back on. At high RPMs voltage is sufficiently regulated to not cause a problem in the circuit. Do you have an old-style VR with resistors on the back? is a good question to ask if you can't trace it to bad ground or bad connection.
Above image is what the back of my old VR looked like (I believe it is original with the car). The new Solid State versions don't look like this.
Most likely, this is not your problem, but at least I stuck to the "radio cuts out" theme. Hope this wasn't too much of a thread high-jack!