Bristol City's Taylor, 19, is an England youth international and joins on loan until the end of the season.
Brown, 25, brother of Blackburn and former Manchester United defender Wes, spent the 2015-16 campaign on loan at the Shakers and was released by Sheffield United on Friday.
Fulham's Burgess is a 21-year-old centre-back or left-back who will stay for the rest of the season on loan.
He spent the first half of the season at Bury's League One rivals Oldham, making 31 appearances in all competitions.
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Bury have signed defenders Taylor Moore, Reece Brown and Cameron Burgess.