Corporate Captain reported a low altitude alert; issued by ATC; while on an instrument approach. Flight crew executed a missed approach and landed.

Narrative

I was flying a Hawker HS-125 into ZZZ. Weather was reporting OVC 500 feet and roughly 2 SM. ILS Runway X was in use. This particular approach has a mandatory altitude of 1500 ft MSL at the waypoint ZZZZZ. My copilot set up and confirmed the approach in the FMS. We armed the VNAV and NAV when cleared for the approach. We discussed not arming the approach; as we feared it would try to go back up and capture the ILS glidepath that was above us and not meet the 1500 required altitude. As the VNAV was active and descending; I noticed that it did not level off at 1500 feet and began descending prior to ZZZZZ. I disengaged the autopilot and asked my copilot to set 1500 feet in the altitude selector and arm the approach; as I was handflying in IMC and it was a critical phase of flight. I then initiated a descent with the Flight Director; which I thought was following the glideslope. My copilot and I realized that this was an incorrect indication and we both decided to initiate a missed approach. At this time; I believe I was at roughly 900-1;000 feet MSL. ATC issued us a low altitude warning as we initiated the go-around. We followed the published missed briefly and then were given alternate missed approach instructions. When we did the approach again; we manually captured the 1500 feet at ZZZZZ and then armed approach mode; which autotuned the ILS.