I had a friend who raced on a Lambert frame in the early 80's, and he was no slouch about bikes. It was a great frame, and he was proud of it. The BB was changed over to something reliable. I don't remember the details, but I'm pretty sure you'll find a standard sealed BB fits the threads just fine.
No doubt someone will tell us more about the frames; most were fillet brazed chrome-moly tubing, but I have also heard of lugged Lambert frames.
The Lambert spindle is not a regular square taper; the Lambert spindle is a little thinner and, as I recall, it doesn't taper at all! You can file a taper into the crank arms, though, and fit them onto a tapered spindle. It'll give you the original look, very much like a TA crank, but not a very stiff one.
There are no threads! It is a pressed in BB using readily available #6003 sealed bearings. The spindle had external snap rings and would occasionally (well, eventually) break at the snap ring groove. I had good luck replacing these with Klein spindles BITD, and I understand Phil Wood has appropriate spindles these days.
Do not discard the crank bolts if you keep the spindle and cranks. They are British Cycle Thread 5/16 X 26, which in the USA is rare and bizarre. Do not try to use the old bolts in any new spindle you may purchase.
So if you feel the need to make a change, check for an old Klein dealer to see if they have spindles. You can get just about any length you want, tapers for most cranks, and if you don't ride the thing underwater, you will do the job ONCE. I used to spend ~$15 for a genuine Klein spindle ~15 years ago (I think there was a professional discount--it was the shop down the street from mine). I hear that Phil gets ~$80 for a spindle these days.