Despite the sabre-rattling there's only so much that the BBC and Moffat could actually do as almost all of the original narrative is now in the public domain in the US. Any similarities that exist between the two shows that are derived from the source material would be fair game.
Based on the promo video, the backstories are different in Sherlock
, and I wouldn't be surprised if Watson's military background was dropped in Elementary
simply to avoid a problem with Sherlock
. I also wouldn't be surprised if Elementary
's Watson wasn't a writer for the same reason.