During the attack the turbolift system was overworked and backed up with requests.
At the same time, there is only one way off the bridge. The computer knows this and that the ship is at red alert. It also knows the captain and most of the command crew are on the bridge.
When Kirk signals he is leaving the bridge, his turbolift has to be prioritized despite them all being occupied and the nearest lift is rerouted.
It just so happens it was the lift the Scotty was using to take his nephew to sickbay with.