The 24-year-old spent last season on loan with the Brewers keeping 12 clean sheets from 43 appearances for the League One side.
Manager Jake Buxton told the club website:
"We're delighted to get our number one target.
"The goalkeeping department and the rest of the squad is very competitive."
The Brewers have also brought in left-back Owen Gallacher, 21, who has spent the last two years as part of Nottingham Forest's Under-23 set-up.
Gallacher, who can also play as a winger, had a loan spell with Harrogate Town last season and had been on trial with the Brewers.