First and most important:

Determine whether the inside of the rims is straight or has hooks at the outer edge to retain the tire. Straight-sided rims limit the maximum safe pressure. Having seen tires blow off straight-sided rims, I ignore the "inflate to" instruction on the tire and pump them to no more than 65 lbs.

Second:

The widest tires currently available to fit 27 x 1 1/4" rimes are almost certainly 27 x 1 3/8". Among the few tire models available in that size, some actually measure only a small fraction of an inch wider than the standard 27 x 1 1/4" tire. Panaracer and Continental tires are usually mentioned as measuring a true 27 x 1 3/8".