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
> Still, how long would it take someone to work it out empirically enough to have confidence that there wasn’t just some bug somewhere?
They probably would just assume that it's a bug at first, since "our clock is slightly broken" is far more likely than "we've discovered some cool new physics". But I don't think that that would be an issue here, since even if they thought it was a bug, they'd still try to work around it.
> Maybe they would mothball the GPS project for not working before figuring it out?
I doubt it, since if you think about all the possible issues after sending a complex untested satellite to space, a 40μs clock discrepancy is fairly minor.