Wilcox moved to Dean Court in December to act as No.2 to Bradbury, who was sacked on Monday, and will find out his future in two weeks' time.
It was Wilcox's first role in football that was not alongside Brian Laws, having worked as his assistant at the Iron - where he finished his playing career - as well as Sheffield Wednesday and Burnley.
Another former Iron player, Paul Groves, was put in charge on a caretaker basis.
"I think the players would be disappointed in respect of Lee and Russ being good people, but they know that these things happen in football," said Groves.
"They've got jobs to do themselves and they have to get on with their jobs. We are preparing the players in the best way that we can in the limited amount of time that we have."