> 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.