I would be quite nervous if I thought I was riding on Titanic-era rails - the quality of steel back then wasn't anywhere near what it is now
Does the safety case for any individual line take into account rail age and consequent brittleness? Or are the only surviving rails of that era in locations where slow speed is absolutely mandated (ends of terminating platforms, sidings) and a fracture then unlikely to even cause damage?
It's unlikely you've travelled at any speed over much rail that old, as it is likely to be stuff like 85 pound bullhead and confined to shunt necks and under buffer stops (see below).
There is a pogrom of sorts to remove pre-1974 rail where high speeds / high cant deficiency are concerned, as this was the date (roughly) rail production changed from Ingot to Concast (continuous casting), the former generally producing rail which has higher levels of slag inclusion, gas bubbles, etc.
Length of service isn't really a factor, whereas wheel burns, RCF, chair gall, head wear, piping, side wear and corrosion are more likely to trigger replacement. That said, as tonnages, axle loads and speeds all increase the older, lighter sections no longer have enough strength as a beam to resist the static and dynamic loadings.
The oldest piece of infrastructure I've ever found on GEOGIS was 1923 (but GEOGIS isn't far off guesswork sometimes) and on site was (though it's probably still there) an 85RBS bent-rail buffer stop on 1903 chairs at one end of a loop on the West Coast Main Line.