alt Hacker NewsRelativity is required to explain the time dilation, but it's not needed to notice that the GPS clocks run 40µs too fast. It wouldn't take too long for them to measure enough data to create an empirical skew-versus-altitude table, which should then be enough for them to correct the time skew, even without understanding what was going on.
I think this is a difficult counterfactual to assess, we didn’t develop clock accurate enough to notice the time dilation until 50 plus years after the theory of relativity was published.
Still, how long would it take someone to work it out empirically enough to have confidence that there wasn’t just some bug somewhere? Maybe they would mothball the GPS project for not working before figuring it out? It’s hard to know, 50 years in an alternate timeline allows for a lot of domino effect