General aviation instructor pilot reported a near miss with another aircraft in the traffic pattern. Both aircraft were instructed by ATC to go around and offset to avoid the traffic conflict.

Narrative

I was instructing a COM student in the traffic pattern at BED. While departing from a touch and go; we requested a short approach and were informed by ATC to continue upwind to provide us with this approach. Eventually on the downwind we were asked to locate traffic on a 1 mile final and cleared to land #2. We spotted an aircraft on short final; informed tower we had traffic in sight and immediately turned base - assuming our short approach was still valid. Checking my tablet I noticed an a/c which I assumed to be the a/c on short final. Unfortunately; there was another aircraft further away from BED on final which we did not see. The other a/c questioned BED tower regarding us and was still instructed to land. Eventually the other a/c announced they were going around; and we were also instructed to go around - 'isn't going to work out and offset to the left.' The other a/c was also instructed to offset left but informed tower that this would put him in conflict with us and they offset to the right.We should have questioned tower whether our short approach was still approved and done a more thorough job of checking final for additional aircraft. ATC should have instructed us to go around prior to the other a/c advising them that they were going around. ATC should have instructed the two a/c to offset to opposite sides of the runway. A number of voices were heard acting as ATC giving concern to the nature of both pilot and ATC training at BED.