I don't think there's any good, in-universe explanation - I'd hate to think that the ship's chief engineer would react in that manner under any circumstances. He's had many years of emergencies before this event, and I just don't see him doing what's shown here; even though it's his nephew who's injured.
Chief engineer or no, he's human. Kirk fell in the middle of his bridge and started sobbing after he learned David had been killed. He also ran toward the power conduit where Spock was dying even though opening the door would have flooded the entire engineering section with lethal radiation. He was stopped only because Scotty and McCoy held him back. Didn't he have years of experience, too?
What Scotty did may not make sense. But that's the point. People react strangely when they're in shock. When that shock is precipitated by the death of a close friend or family member, any job training or experience goes right out the window.
That still doesn't work for me. Kirk's reaction on the bridge, and in engineering were both immediate reactions to unforseen circumstances. Scotty's reaction to bringing Preston's body to the bridge is considered and time-consuming - and by going from Engineering to the Bridge, he had to pass Sick Bay. Or he was at least closer to it in transit than he was by the time he reached the bridge.
But no matter, we're debating (IMO) an artistic decision. Interestingly and coincidently, I've been reading the novelization of The Wrath of Khan, and just read through the scene in question. I've got to say I much prefer Vonda McIntyre's treatment to the filmed version. In the book she has Kirk and Spock going to Engineering immediately after the battle - to assess damage. They're met at the turbolift door in Engineering by Scotty with Peter in his arms. Scotty says has not been able to reach Dr. McCoy and he must get Peter to Sick Bay. Spock takes Peter from Scotty's arms and they all head directly to Sick Bay. Well written, and still dramatic - I wish this is how the scene was filmed for the movie. I don't know if this was the author's interpretation, or if this part was in the shooting script, but changed. Does anyone know?