I have had this problem before I think. If you're not getting any sort of dmesg output to the screen, then the kernel is completely unable to calibrate itself for the processor. I also assume that the reason your HD swapping between boxen is probably because the Cyrix doesn't have a bootable CDROM, or doesn't have a CDROM at all.
If RH's basic boot kernel doesn't work, and the reason it doesn't is that the cyrix, technically a P1, really doesn't correspond to all of the i586 architecture standards, you can poke around for more RH boot kernels, or make one of your own.
You can choose a different distro, offhand I'm preferential to Slackware which compiles its kernels for the minimal architecture (which makes them slower), assuming that the first thing you're going to do with the running machine is recompile the kernel specifically for that machine's hardware. Also Slack has about 20+ bootdisk images to play with.
Since both the P1 and the Cyrix are Socket 5, if you live anywhere near a place that sells used parts, you can probably pick up a P1 for the ballpark of $5-10. This I wouldn't normally recommend, but you seem rather gut-the-machine savvy if you HD swapping. Who knows, your board may support up to 200, although I doubt it.
This is all assuming that the bizarre-ness of the Cyrix is the problem. Offhand it could be a video card issue (doubtfully), or a RAM problem (maybe, but still doubtful, especially if this thing has had a running OS recently). There is some other stuff, but most of it is so out there as to be lottery unlikely.