Will 27 x 1 1/4” tires be too wide for an old bike that came with 27 x 1 1/8” tires?

In lieu of more information about the bike, the best answer is probably not, but maybe. You can find some examples of bikes that came with 27x1-1/8" that also max out at that size, but they are not very populous relative to ones that could fit a little bigger tire. As always with questions like this, it all depends on the exact tires and rims in question anyway, because there is variance in the actual size among tires that bear the same nominal size, and among inflated width among different rim widths.

Also since you have the old tires, you're in a better position to figure it out than us. Mount them on the new rims and install on the bike. You probably have choices in dropout slot position that can affect clearance. Measure the inflated width and height. Most 27x1-1/8" tires run pretty true, i.e. 28-29mm wide on the sort of rim they're intended for, approximate inner width 15ish and outer 20ish, or a little bit wider and shorter if your new rims are chunkier, like if they're more replacement wheels for bike boom 10-speeds with their 22-23mmish outers, to which a more typical 27x1-1/4 also runs pretty faithful and lands about 32-33mm wide and tall, or narrower and taller on a slightly roadier rim. From there you can make the inferences needed to figure out if the "average" i.e. pretty true to size 27x1-1/4 will fit.