The time does not change in Galilean transformations precisely because there is a global time that can be defined.
We now summarize this discussion. The key points are: time has to be realizable physically, i.e. we have to construct some clock that tells us what time means. Relating times at different points requires a well defined, physically realizable method of synchronization. In order to construct a time that is well defined everywhere, we need an infinite number of observers with synchronized clocks. Newtonian mechanics requires the assumption that light travels infinitely fast, which is a physical assumption about how the world works.
My hope is that you see the key roles the requirement of physical clocks and the assumption about the speed of light play. If you think about time as a family of physical clocks ticking, then the fact that time changes between Newtonian mechanics and relativity is not so mysterious or surprsing. Eventually, you want to get to the point that of course if light doesn't travel infinitely fast then the Galilean transformations can't hold. Of course if we measure the speed of light to be the same for all observers Newtonian concept of time can't quite be right. Of course if we take the limit that the speed of light be large we translate from relativity back to Newtonian mechanics.
Next set: space, coordinate systems, and distances.
Sets after: metrics, Lorentz transformations
Then: black holes, entropy, black hole entropy
This set has been all conceptual. As we develop the conceptual underpinnings we will move on to more sophisticated technical aspects.
Everyone: go find extra free reading on this topic. If you find something you like, put it in the discussion of this note for your fellow Brillianteers.