How often does it blink? How long is it on and off? Fraction of a second or always 1 second?
Is it regular or irregular?
If the led lights continuously then the voltage on the input is very low or 0. Actually is seems continuously. As the measure voltage is very low, so is the switch-on/off delay. (miliseconds) Either you have the wrong pin of the potentiometer connected to the analog input (Gnd/0v to the A0 pin), or there is no voltage across the entire potentiometer (incorrectly wired/broken wires), or it’s internals are broken. Pictures could identify the first 2 possibilities. Measuring resistance through the pins of the potentiometer with multimeter (essential tool) would rule out the latter.