After lots of research, I decided to replace the original cottered cranks on my Raleigh Grand Prix with Velo Orange's threadless BB and Grand Cru compact double. I went with a 118-mm spindle, since that's what was recommended with this crankset.

Removing the old coffers actually proved a snap -- done in about 10 mins with a c-clamp.
But the new BB was a battle because the Park Tool lockring wrench constantly slips out of the not-very-deep notches on the outer rings of the BB. Finally though, after much elbow grease and a nasty looking gash in the previously perfect paint on the BB shell, it looked seated. I put on the drive side crank and immediately knew I had a problem. The arm is close to the chain stay, and the chainring is not even clearing the drive side cup. I removed the BB and see now that the threading on the non-drive side is stripped (the threads don't engage the BB shell but rather, the expanding cup things) and this was not totally tightened.

This leaves me at an impasse. Do I:

1) Order another Velo BB in a different spindle length. The company offers 122 and 127 lengths.
2) Order another in the same size and try not to strip another.
3) Be thankful I didn't totally f-up the frame (it's my father's and this worth a lot more to me than your average $35 GP), grease the threads, and reinstall the heavy old cottered cranks.
4) something else.

If you want a square taper BB you can't beat Phil Wood. Last time I looked they had a huge range of choices of widths and tapers.

Assuming it's 68 British thread any of the modern cup BB's will screw in. There's a wide variety of compact cranks that work with exo style BB's, and they are a snap to shim, especially if you use a pinch bolt type crank (Dura Ace). I've built several old steel frames like this.

What did you end up doing? Was the bb spindle the right length and were you doing a singlespeed?

I am doing this same thing on a 70s Raleigh Sprite. I've got a sugino xd crankset that I intend to install as a single speed with the velo orange grand cru threadless bottom bracket. My only real question is which spindle length to buy to achieve the correct chain line. The crankset has holes/threads for a triple but I will only use one and ideally the outermost spot for the chainring. 71mm bb. Any tips?

Some of those old Raleighs used 1-3/8" x 26tpi bottom bracket thread, which may be why the OP elected to use the VO threadless cartridge. I doubt that the VO cartridge stripped the BB shell; rather, I suspect that the OP may be seeing aluminum debris from the cartridge plugging the threads. A stiff wire brush or sharp pick may be able to clean them out. If the OP can get the threads clean, Phil does (or did, anyway) offer 1-3/8" x 26tpi lockrings for their cartridge bottom bracket set. Seems like overkill, but if it gets the bike on the road it may be worth it to the OP. Otherwise, if the original cups are in reasonable condition, they can be used with an appropriate square taper spindle to mount a standard cotterless crank.