I have a vintage bike which requires a new crankset (and bottom bracket). Long story short, pedals fell off the old cranks, I bought new cranks and those cranks didn't fit the old square tapered bottom bracket. So, I am trying to get a new matching BB for the new cranks.

First, the cranks I got are the Sunlite Crankset, Cotterless Steel, 52/40 x 170mm that is square tapered

It didn't fit into the old BB, which was also square tapered, as the spindles were too big. So, i need a new BB.

Second, the BB shell is the standard 68mm. And, the spindle on the old BB is 127mm. That, having been said, what BB should I get? I was looking at the Shimano UN55 68x127 square taper. Would that work?

For what its worth, the old BB was manufactured by Tange and had the following numbers 1.37x24T inscribed on the lockring. And in the middle of the spindle (it's not a sealed BB), it has the following marking: "3S3"

So, any recommendations on a BB to fit those new replacement cranks?