I'm guessing that the small cog on the Regina has a slight warp in it or has been cross threaded into the 2nd cog. The chain is most likely not the culprit as others have stated.

I've seen this warp occur most frequently on the old Regina and Atom 5 speed freewheels where all the cogs thread onto the body or another cog (which the smallest Regina cog does). Because the cogs can be so stubborn to remove for cleaning or re-gearing, a significant amount of force can be applied with a chain-whip. If that force is not on the same plane as the cog, the warp can occur. It doesn't take much warp to cause the chain to skip.

Can you take pictures of the Regina from the side while it is placed on a flat surface (such as a kitchen counter)? Take at least 4 pictures as you rotate it 90 degrees for each one.