Yep, this is a strange one. There's no way 'Common' would be the same on each world. I think most folk hand-waive it.
Not only that, but several D&D worlds have archaic versions of the current Common tongue, like the Thorass language. So if "Common" is truly quantifiable with English, then somehow on every campaign world, the language evolved into it's current form, in very different ways.
English evolved due to contact with other cultures over the course of Britain's history- but there simply might not be analogues for all these cultures in a given D&D world, or if there are, the majority of Common speakers might not have come into contact with those cultures at all!
But somehow we get to the same end point. Talk about divergent evolution!
Where this really gets interesting is that English is continuing to evolve (or mutate, or devolve, depending on one's point of view) as we speak, with words acquiring colloquial meanings very different, and in some cases, completely opposite from their intended meaning, the introduction of slang or made-up words ("All words are made up") to the lexicon.
Which makes it entirely unlikely that a Common speaker will be comprehensible to an English speaker as time progresses. The only reason the language hasn't broken apart already into similar but distinct sub-dialects is only because of how "small" our world is- people from across the world are in continuous contact with each other, and for the purposes of communication, keeping languages from diverging too much is somewhat necessary.
Most D&D worlds often have communities separated by large distances, and contact with one another is far less frequent, when (barring magic), the fastest way to communicate is by horse, ship, or carrier bird!
If anything, the existence of roving bands of adventurers* who travel far and wide are probably the biggest contributor to keeping Common speakers in contact with one another- upon entering a new town, an adventurer can probably find themselves being pressed for information about the exterior world.
*Or Bards, or merchants, but given how dangerous these worlds can be, they are often associated with, if not adventurers themselves!
Of course D&D often has an easy out. Things are the way they are because the Gods (or cabals of power NPC's) want it this way, and exert significant influence to ensure this fact.
