How do I find the right size of bottom bracket?
It would still be helpful if you posted a photo, but I think I can answer without it! 

The first major dimension of a bottom bracket is its width. If you have a metric ruler or tape, simply place it underneath the bottom bracket shell. There aren't a bunch of standards super close together so you'll easily get this one right. 68mm is really common and would make your project very easy to complete, but it's ok if it's different.

Like @Criggie said, the 1.37x24T measures the threading. Threads are the next most important dimension, and yours is standard English which is very easy to replace. 

Like I said in a comment, your old spindle was probably asymmetrical, but new ones are typically symmetrical. To get your correct spindle length, you need to measure from the center of your spindle to the outer edge of the drive (or right) side. Double that number and you'll have the correct new spindle size. You'll probably want a pair of calipers for this, but the $5 ones at the hardware store are more than sufficient. 

I did a quick google and your bottom bracket is probably a Sugino. All I did was type in "SCM 3UB bottom bracket" to get there. This is a Japanese part that is likely NJS spec'd (criteria for use in Japanese track racing) so it very well could have an ISO taper and not a JIS. See here from Sheldon to understand these terms. If your measurements are perfect and your crank won't slide on to the right place, this is the next culprit. If this BB is ISO, and your replacement BB was NJS, then your crank arm was probably about 4mm off from where it needed to be. 

Did you already get the fixed cup out of the bottom bracket shell? If you didn't, I'd just take it to the LBS for help. They can do all of these measurements (including checking to see if it's NJS or ISO) for you, and that way you only buy one bottom bracket- the one that fits.

Good luck!