I've witnessed first-hand an A320 pushed with no bypass pin inserted, though it was done completely in error. The gate lead failed to notice that it was not inserted before the push.

From the ground, nothing seemed to appear or behave any differently (obviously nothing to go by, though.)

I work on the ramp as well, so I can't say with any definitive-ness, but I *think* that all the bypass pin does on the A320 family (and probably most other aircraft) is disconnect nose gear steering from the tiller in the cockpit, as a safeguard from the captain trying to turn the wheels while the nose gear is still connected to the towbar and tug, which would obviously cause damage and be dangerous for the ground crew.

I've never heard of your circumstance happening. That would be against company policy at my airline, and I imagine at most others.