2038 problems have already started happening in niche cases. I expect things will pick up in pace in a couple of years because 10 years is that sort of period of time that people like to post-date future events by.

That might be enough to scare a few of the hold-outs. Then 2033 will be the next scare and reminder, because five years, etc.

Then probably every year after that until the deadline hits. By that point, the remaining few will be using faketime or something like it to eke out a few more years from whatever ancient hardware they're still running that is too expensive to replace.

Fun fact, and possible hint: Setting the date back 28 years on such hardware could work in a pinch, since the calendar from 2010 to 2037 is identical to that from 2038 to 2065. All weekdays and leap days fall on the same dates. (Easter and other moveable feasts don't, however.)