Would Fate Stay Night count? They were done by Studio Deen and then Ufotable.
In the case of Unlimited Blade Works, I would.
Other reboots done by different animation studios I can think of... -Casshern- The original 70s anime was done by Tatsunoko Productions. The reboot done with Casshern Sins was done by Madhouse. -Raideen- The original 80s mecha anime was done by Sunrise. The 2007 remake was done by Production IG. -Berserk- The 90s TV anime was done by OLM. The film trilogy reboot was done by Studio 4C. -Jojo's Bizarre Adventure- The earlier film and OVA adaptations of varying arcs in the series were done by Studio APPP. The current TV anime being done since 2012 are done by David Production. -Negima- The 2005 TV anime adaptation was done by Xebec, while the 2006 reboot for it was done by Shaft. -Evangelion- This one might actually come as a shocker to Eva fans. While the TV anime and End of Eva were done by Gainax, another studio named Studio Khara is handling production for the Rebuild of Eva film series.
Someday I’ll be gone To somewhere that we belong And God has never played his role 'Cause I’m the one who saves my soul It’s a perfect world we’re longing for