Who originally sang the carol 'Deck the Halls'?

William Byrd? Thomas Talis? Queen Elisabeth's court capelmeister?

Unlikely, actually. The tune is Welsh in origin, Nos Galan from the 16th century, but the words are 19th century American! Mozart used the tune in the 1770s (but he was not averse to pinching a good tune from anywhere). But I've not been able to find the author of the words. Very widely reported as being American, and not in repititions of a Wiki article, either. Probably dates from the times of Dickens and his tours - he spread a lot of the Christmas cheer type of stuff that we now regard as traditional.

J.P. McCaskey is sometimes credited with the lyrics of Deck the Halls but he only edited the Franklin Square Song Collection in which the lyrics were first published.

The first publication date of Deck the Halls is 1881. The author is unknown but the words are said to originate in America.

I wouldn't think Mozart would have used it. He was more into French and Italian melodies. The English, Welsh and Scottish tunes were largely arranged by his good friend Haydn and Beethoven.

"We do know that by the eighteenth century the melody had traveled far enough to have been used by the great master Wolfgang Amadeus Mozart (1756-1791) in a duet for violin and piano." (William Studwell, 'The Christmas Carol Reader')
https://books.google.com/books?id=IPqjW-AFAsUC&pg=PA144&lpg=PA144

"In 1788, Austrian composer Wolfgang Amadeus Mozart wove the tune into his Sonata in G for violin and keyboard K. 301 (NMA K. 293a), commonly identified as "Sonata No. 18." Mozart may have heard this Welsh folk song during his boyhood visits to England."
https://en.wikipedia.org/wiki/Deck_the_Halls