The capacitance of the solderless breadboard may be causing your issues. This is a finicky circuit. The creators are proud that they can make this work with only a few parts, but real circuits have more parts for a reason, to make them work reliably.
I was curious so I made the circuit. I used a BC546, basically the same as your transistor. My coil is 26 turns of 30 AWG solid wire, about 2 cm in diameter. I soldered the parts in mid-air. I started with a 1k base resistor. I set the current limit on my supply to 0.1 A, the rating of the transistor. It did not work. I lowered the base resistor to 100 ohms, it started working.
The supply current limits at 0.1A, ~1V!
The coils need to be within about 2 cm, with their axis (holes) aligned for the LED to light. Use a red, yellow, or orange LED since these have low voltage drops.
So, be sure that your coils are small. Don't use a solderless breadboard. Use a small PS voltage with the current limit set to 0.1A.
Channel 1 is the base voltage, channel 2 is the collector. Emitter is ground. The transistor is being stressed, the Vebo is being exceeded.
simulate this circuit – Schematic created using CircuitLab
