Anderson, who is yet to make a senior appearance for Burnley, played 18 games on loan at the Spireites last term.
The 22-year-old has previously had temporary spells at Carlisle, Lincoln City, Halifax and Hyde United.
Republic of Ireland under-21 international Wilkinson, 21, is yet to score in 11 games for Bolton.
He had loan spells with League Two clubs Newport County and Portsmouth last season, scoring one goal in 13 appearances.
Meanwhile, Chesterfield's teenage forward Curtis Morrison has been loaned to Gainsborough Trinity for a month.
Find all the latest football transfers on our dedicated page.
Chesterfield have signed Burnley centre-back Tom Anderson on loan until the end of the season and Bolton striker Conor Wilkinson until January.