We know he died, as he spoke as though he was going to, and his arm fell limp by his side with Nanaki frantically nudging him but getting no response.

And notice how nervous Nanaki and out of character Nanaki was when he's lying to the others:

Cloud
"Red XIII... Is Bugenhagen...?"

Red XIII
"......"
"Grandpa told me that he's going away again!"

Cloud
"Away...?"

Red XIII
"It seems like he really liked that Airship."
"He told me he couldn't stay and just zipped out of here!"
"He even gave me a gift, look!"

Red XIII
"Wait a minute. Aren't you supposed to give gifts AFTER you
come back?"
"Ha ha...... Grandpa sure is strange! Haa... ha ha...... ha......"

Cloud
"Hmm... So maybe we'll run into him again somwhere."

Red XIII
"Yeah... maybe..."
"Thanks, Cloud......"
He lied to them for the sake of their feelings, the same as Vincent lied to Lucrecia about Sephiroth.