We got to see the uniform switch take place on DS9 when they went to the First Contact
-style uniforms mid-season. In fact, you can see the two uniforms side-by-side in DS9: In Purgatory's Shadow, when they find the real Bashir on the Jem'Hadar internment camp. He's still wearing the DS9 style uniform, implying that he was kidnapped before everyone else on DS9 switched over to the new jackets.
Me personally, I had zero problems with the Generations uniforms switcheroo. As others stated, modern militaries have multiple approved uniforms. Just because we never saw them wear the jumpsuits on TNG doesn't mean that wasn't an option available for them.