日本国語大辞典 says:
近世「だれ」と変化したが、「随筆・松屋筆記‐五五・三」「徒然慰草‐一・二二段」に「今云あの人たちのもの申さるることばをきくに或は『たれ』といふことを『た』の字をにごり〈略〉かやうのかたこといくらといふかずをしらず丸か耳にさへをかしく伝ればこれをいにしへの人たちにきかせ奉らばいかばかりかなしくおぼしめされん」とあり、近世後期からの現象と思われる。現代では、「だれ」が一般的であるが、主に文語脈の中では、「たれ」ともいう。
Judging from 松屋筆記 is written in 1818-1845, and 徒然慰草 is some point before 1650, the current pronunciation gradually replaced it during the Edo period. 誰 is etymologically from voiceless た, but assumably changed into だれ by analogy to ど-series question words.
Is this correct for the time period?
Well, it's "correct" for the story's background which seems to be late 12th century, but considering that the story is written consistently in today's Japanese, I'm not sure if it's faithful to the author's intention. As I noted in my comment, I couldn't find other recitation saying たれ on YouTube.