Yet when I put the same bottom bracket axle in my modern bike, which has the same bottom bracket shell width and the same chainstay length, all works perfectly. Can you suggest a remedy?
This sounds very much like frame misalignment. If the chainwheels and axle work on the new frame, they should do the same on the old.
One explanation is that, to get the required spacing, the frame builder pulled the stays on one side (probably the gear side) further out than the other – or even pulled them all the way over on one side. In other words, the rear wheel may be offset from the cycle’s centre line.
A quick check is to remove the rear wheel and run a length of string from the inside of one rear dropout, around the head tube and back to the inside of the other rear dropout in a symmetrical arrangement. If the dropouts are evenly spaced from the frame centre line, the string will be equidistant from the sides of the seat tube. Should this prove to be the case, the rear end should be tweaked accordingly.
Cycle’s Technical Editor