Fractional pilot reported receiving a low altitude alert from ATC while on the ILS approach and experiencing localizer fluctuations. Pilot corrected altitude and continued approach.

Narrative

Low altitude alert. While flying the ILS 6 at TEB we experienced the LOC fluctuating 1-1.5 dots left and right outside of DANDY. Conditions were VFR and the airport was in sight. The Autopilot was on and it started to chase the needles. This monopolized my attention and led to a degradation of SA. I almost missed the 1500 altitude restriction at DANDY; but the second in command (SIC) reminded when I failed to start the descent. At this point I disengaged the Autopilot and initiated a descent. Crossing DANDY the received a 'low altitude alert' from the Tower and I observed the altitude at 1400. I corrected the altitude and continued the approach in visual conditions. Circling to land Runway 1. I did not receive a brasher warning from TEB Tower but I was expecting it. After landing I observed men and equipment working on Runway 6 directly in front of the LOC antenna. We reported the issues with the LOC signal to Ground Control. Another plane overheard our transmission and reported to Tower that they had also experienced fluctuations. The LOC frequency has very recently been changed at TEB as well.