I think it'll be a "we'll see" situation. This was the main concern for y2k. And I don't doubt there's some stuff that was partially patched from y2k still around that is still using string dates.

But the vast majority of software now works with timestamps and of course some things will need work. But with y2k the vast majority of business software needed changing. I think in this case the vast majority will be working correctly already and it'll be the job of developers (probably in a panic less than a year before as is the custom) too catch the few outliers and yes some will escape through the cracks. But that was also the case last time round too.