Thanks, guys, but there's something still missing. Looking at the IAP, the DME is quite clearly co-located with the localizer transmitter, and is down near the far end of the runway. The localizer frequency is indeed common to both runways, but the equipment is not - they just turn the one associated with the active runway on at any given time. You need to look at the EGPK IAP to see this.
I can accept that the DME can be tricked into displaying a shorter distance than the physical distance between the airborne equipment and the ground satation - though even that is quite a trick, because it will require speeding up the round trip time of the signal.
Usually on the missed approach, you just go straight ahead to some specified altitude and then make a turn, but Prestwick specifies a turn at 0.7 DME. I'd be really interested to know how a DME that is tricked into displaying distance to the approach threshold on final can be tricked into displaying distance to the departure threshold on the missed approach. If it is set to show, say, 1.5 miles less than true distance on final, then I'd expect it to show 1.5 miles less than true distance at the departure end as well. This would work if the DME ground equipment was located exactly half way down the runway, as suggested by one of the responders, but the IAP shows quite clearly that this is not the case. I guess it will take someone who is responsible for setting up the ground equipment to convince me what is going on here.