While complying with the commands of a TCAS RA with and unknown aircraft the flight crew of an HS-125 complied with a changed heading clearance from ATC which resulted in an NMAC with a third aircraft.

Narrative

We were descending on a VFR day into BAK; number three for the approach to Runway 23. While being vectored to the ILS we received an RA to descend. After reducing the thrust levers to idle and verifying the green area on the VSI; I executed a descent from our assigned altitude of 4;000 MSL to 3;000 MSL. During this time the Captain advised ATC that we were responding to a RA. ATC then issued a vector to a heading of 320 while in the descent. While making the turn I spotted a light single engine T- tail trainer. The relative position of the second aircraft was on a 320 heading. This would have put the aircraft at our 12 o'clock position and at the same altitude. I stopped the turn prior to reaching the 320 degree heading. The two aircraft passed within approximately 100 yards of one another. After clear of both conflicts; we visual conducted a visual approach to Runway 23 with no further incident. I do not believe there was any way to prevent the chain of events leading to this situation. The second aircraft apparently did not have an operating transponder.