I don't see why anyone would be offended by either phrase, but "From A to B" and "From A to Z" do have different meanings.
"From A to B" usually refers to a journey, where A is the start point and B is the destination. So, "Communicating All The Way From A to B" sounds like "Going from your start point to your end point, and communicating in the process." It might be suitable if you were trying to sell a mobile phone which had a feature of keeping a call open during a train tunnel, or something - you're saying "You will be able to communicate over your entire journey".
From one's starting point to one's destination.
‘most road atlases will get you from A to B’
"From A to Z" is a reference to a dictionary, and is intended to indicate thoroughness - ie that you will go all the way through the dictionary from start (words beginning with A) to finish (words beginning with Z). I'm not sure where you might use "Communicating from A to Z", but what it's implying is that your communication strategy is very thorough, perhaps.
Over the entire range; completely.
‘make sure you understand the subject from A to Z’
Anyway, I don't know the context so I'm just speculating about possible uses, but that's the difference between the two phrases.
As far as I know these phrases are not unique to any variant of English and can be used with AmEng or BrEng or whatever you like.
EDIT: As Davo & Yosef Baskin point out in the comments above, if the context is a range and hence "A to Z" is expected, and you say "A to B" instead, it could be taken as a deliberate statement that the range is very limited, rather than being thorough.