If $f:X\to S$ is a universal homeomorphism, is $f':X\times_S X\to X$ always a nil-immersion? This seems to be easy, yet possibly I miss something. Should I give references to this fact in a paper?
In general, the answer is no. As Brian Conrad points out in the comments above, purely inseparable field extensions do not have this property. However (and maybe this is what you were getting at?), it is true that if $X\to S$ is a universal homeomorphism, then the diagonal $X\to X\times_SX$ is a nilimmersion.