If the player has a good ear (and you're not asking them to do really virtuosic playing), then the "embouchure shock" should be fairly minimal.
As long as they're still reading transposed treble clef music, the fingerings will be exactly the same, with the notable exception of french horn (or is that a brass band instrument to start with?). Tuba doesn't read treble clef traditionally, but a treble clef, transposed part could be easily created that would work just fine for the player.
If the player is brand new at brass instrument doubling, then yes, they may need a few days to get used to it, just for the fact that it's different from a cornet. A more experienced doubler can usually switch around at will with no "adjustment period".