Apparently the Romulans had both a double of Picard and a virtual double of Yar for years.
To take us down a path bordered by wild Epileptic Trees for a moment...

Might I suggest that it was capturing/interrogating this alt!Tasha from the future which led to the Romulans taking an interest in this apparently unimportant Starfleet officer named Jean-Luc Picard and deciding to clone him in the first place? Baring in mind that Picard was hardly anyone special at the time the Romulans were supposed to have stolen his DNA. If they had some forward knowledge that Picard would be the captain of the flagship sometime in the future, however... that would actually make their decision to clone him plausible.
(Sure, the alt!Tasha was from another universe entirely, but maybe the Romulans weren't to know that. Maybe they captured this woman claiming to be from the future, and simply assumed that she was from the future of our universe, with the full truth only coming out later.)