HS 125 Series First Officer reported while circling to land from an ILS approach they received a low altitude alert from Tower.

Narrative

We were coming into ZZZ from ZZZ1. We arrived on the ZZZZZ with the belief initially that we would be landing on runway XX. We briefed the approach for runway XX in the descent to the airport. The Captain got the ATIS and said that we would be going to ILS runway XY and that we would be asking for the circle. The knowledge of the runway and approach came in later than I would have wanted on the descent which we were already past ZZZZZ. We set up the minimums for the circle and was given direct ZZZZZ1 for the approach which we followed. We were then told to cross ZZZZZ1 at 3000 and intercept the localizer which we complied. Our last instruction from ATC was maintain 2400 and cleared ILS runway XY. Started our descent to 2400 and we switched and the let ATC know ILS XY and that we would like the circle to runway XZ and we had the field in sight. Tower cleared us for the circle and to land runway XZ. What I believe happened next Is that I asked the Captain to set the minimums to 1560 for the circle which he did; I selected vertical speed and started down to the circling minimums. I am not sure where we started down on the approach but when we did get to 1560 ATC told us they got a low altitude alert; check altimeter setting. We read back altimeter setting and confirmed we had the right setting. Captain noticed that we were a mile past ZZZZZ2 the final approach fix and we were at circling minimums and while being four miles from the airport so we were lower than we needed to be at that distance which I had realized after the callout for the altitude alert. We looked to make sure there were no obstacles that we needed to avoid which there were none visually or on the map and we circled and landed without incident at ZZZ.I believe getting the ATIS sooner would have given time to plan; brief and not rush the decision making process and also give both parties an opportunity to express any concerns and solidify expectations during the approach and give a chance to answer and clarify any questions. I feel that I should have verbally asked to be told when we were crossing the final approach fix and to make sure that I knew what the final approach fix was. Also doing a better job of checking the approach on the PFD Would have given better situational awareness. I also think the Pilot monitoring could say when the next altitude is and verbalize when you can start down. Monitor the progress and let the Pilot flying know when its appropriate to descend especially when the approach has been rushed.I think I let myself get tunnel vision and be overly concerned on making sure to honor the circling minimums that I forgot to make sure that I was at the proper spot on the approach to start the descent to get to circling minimums. Better planning; checking the FMS; having the approach plate up or watching the map on the PFD could have solved this error.Verbalizing; confirming intentions; next steps as well as actions being taken could have been better verbalized and communicated between both parties.