The non-Bethesda studio would have to figure out a way to gain some Elder Scrolls experience.
Since its inception in the early nineties, every Elder Scrolls game has been created by a member of the Bethesda family, whether Bethesda itself or its mobile-focused sister company Vir2L. That might not always be the case however; as Bethesda's Todd Howard says that the studio won't rule out letting an outside studio having a crack at the series.
It's probably not very likely though, as Bethesda's preference is keep development of the Elder Scrolls series as internal as possible. Bethesda has used outside developers for one of its franchises before; the most recent Fallout game, New Vegas was made by Obsidian. Howard said that that was a special case though, with all the conditions lining up just right to make that happen. "New Vegas was unique," he said. "We wanted to do something with Obsidian, we knew we were moving on Elder Scrolls, they had a team available ... But they had experience with the IP, and with Elder Scrolls we don't have that."
There's a world of difference between "not ruling it out" and "we're definitely doing this," but it's a fun to think of what other developers would make of the Elder Scrolls series. Would Crytek's take be gorgeous, but focused on ranged combat, or would BioWare's vision be open in the middle, but less so at the beginning and end? We may never find out, but then again, maybe we will.
Skyrim, the next Bethesda-made installment in The Elder Scrolls, comes out for PC, PS3 and Xbox 360 on November 11th.