What's the time on the LEGO clock tile?
This is Part# 4150px1.

Okay, I didn't call/write LEGO for answers. This is just speculative guesswork (and I should be ashamed).  

First off, sorry, for not giving you time to explain the full relevance of the Disney clocks with regards to your question.  There are other clock patterns besides this pattern, and the "Just-Before-Midnight" Disney pattern. 

So, I was trying to get your connection, until I thought about what you wrote, and what I thought in my mind.

You wrote the Disney clocks all have a different time (11:53).  I thought in my head "Just-Before-Midnight", because in many fairly tales, Midnight, is the magically important event.

So, it got me thinking: If I were a Danish LEGO designer, and I was making toys for school aged children, then what would be a magically important event to a grade schooler? If an adult's time could ever be important to a child, what time would that be? 

Hmm, maybe the few minutes before 3:00pm, the time school lets out and for them, the real magic begins?

So, I'm guessing, 2:57:38pm might be the same "Just-Before-The-Magical-Event" subconsciously embedded in every grade school boy or girl's head, and some random LEGO designer had the insight to try and capture it.