Let's see if we can resolve the matter being discussed by having a closer look at the timeline, provided by the actual audio recording of the DPD radio.
On page 11 of this thread I gave a detailed schedule for the audio recording posted by Bill Brown, by using the actual times of the recording.
The key points are;
57:28:67 Bowley starts his call
58:14:42 Bowley ends his call (being told to stay of the radio)
58:24:35 Ambulance 602 calls Code 5 (en route)
58:41:85 Ambulance 602 calls Code 6 (for wrong location at Jefferson)
58:48:40 Ambulance 602 asks dispatcher for address on Jefferson - Dispatcher replies: 501 Tenth Street
59:02:85 Ambulance 602 calls Code 6 (for arrival at Tippit scene)
59:30:99 Ambulance 602 tries to get attention of the dispatcher by calling "602"
59:40:98 Callaway starts his call
59:42:85 Ambulance 602 tries again to get the attention of the dispatcher by calling "602"
We know for a fact that Callaway had not yet arrived at the scene when Bowley finished his call. We also know that he arrived on the scene shortly before the ambulance did. This means that Callaway must have arrived at the scene between 58:14:42 and 59:02:85.
We also know that Callaway said that when he arrived at the scene, he first went to check on the victim before he went to the police car to use the radio. He made his call at 59:40:98, which is roughly 38 seconds after the Code 6 call by the ambulance, at 59:02:85.
Although this can not be said with 100% certainty, it's highly likely that when Butler made the Code 6 call, he was still driving the ambulance towards the location. The alternative would be that he first stopped the ambulance and then made the Code 6, but that would only mean a loss of possibly valuable seconds.
So, if Butler did indeed make his Code 6 call at 59:02:85, it would have taken him - I assume - another 10 to 15 seconds to stop the car, near the victim, and get out of the ambulance.
Butler told George and Patricia Nash that he went to check on the victim. When she saw that it was a police officer he returned to the ambulance to let the dispatcher know that the victim was a police officer. The audio timeline shows this call (the first unanswered "602") took place at 59:30:99, so roughly 28 seconds after the Code 6 call.
Only 10 seconds later, at 59:40:98, Callaway makes his radio call.
All this justifies the question when exactly was there time for Callaway (and Bowley) to help load Tippit into the ambulance before Callaway made his call?