Wood came through the youth ranks at Grimsby Town and was promoted to the first-team squad midway through the 2008/09 season.
He was named as a substitute once that season, in the 2009/10 campaign Wood made his first team debut against Darlington in September 2009, a game which ended 1-1 with him winning the man of the match award.
In October 2009, Wood signed a four-year professional deal with Grimsby before scoring in May 2010 in the Mariners youth team victory over Shrewsbury in the Midland Floodlit Youth Cup final and thus Wood found himself with the supporters` young player of the year award for the 2010/11 season.
Wood played a huge role in the 2012/13 season, as Grimsby missed out on the FA Trophy at Wembley and also missed out on promotion after losing to Newport County in the play-offs. In 2013, Wood was released by the Mariners after making 113 appearances, scoring once. He would join the Reds for his first spell in May 2013.
He joined the Reds on a one-year deal and started on the opening day of the 2013/14 season against Dartford before scoring his first goal in a defeat to Braintree Town.
Wood featured in all-but one of the fifty games in the 2013/14 season and was named the player of the season and would then sign a two-year contract keeping him at the Impact until 2016.
Wood joined Lincoln at the start of the 2015/16 season for an undisclosed fee after Alfreton`s relegation from the Conference (National League) Premier in the 2014/15 campaign.
He would make forty appearances in the 2015/16 season before playing a crucial role during the Imps promotion and magical Emirates FA Cup run of last season in which Wood played against both Burnley and Arsenal in the competition.
Manager John McDermott was ecstatic to have brought Wood back to the Reds: “I’m very happy to have got Bradley back at the Reds. He’s a fan favourite and he signs for the 2017/18 season. He contributed so well when he was last here and I’m delighted to have him back.
“He is a full-back but can play centre mid when required and he can go either right or left back. He had a wonderful season last year winning the league with Lincoln and he kept in contact with us. He loves his football and we know each other personally, it was great.”
“Our style of football suits him and he likes the style, he is a player who knows how to win games consistently, back to back. He knows what it takes to be successful and is a huge asset for us. I want those characters and he’s a leader, plays with his heart on his sleeve and I’m delighted to have him back at Alfreton”.