Worcestershire are also looking to bring in a director of cricket, with former captain Joe Leach - who is currently still on the playing staff - among those interviewed for the role.
"On behalf of the Board, it's a pleasure to welcome Alan and Kadeer into their new roles," Paul Pridgeon, Worcestershire's cricket steering group chair, said. "They both have a deep understanding of the club and are amongst the most respected and progressive coaches around, having excelled in their coaching roles here and in previous roles.
"They deserve their chance to excel and build on our impressive foundations, and they will be further assisted by other important hires over the coming months. We wish them well."
Gidman stepped down after four years in charge, with Worcestershire failing in their promotion push in the Championship and finishing bottom of their group in both the Vitality Blast and Royal London Cup.
Richardson and Kadeer will take charge with immediate effect, bringing the professional and academy squads closer together with the aim of "accelerating the development of younger players" over the winter.
Richardson said: "I've been involved in the club for a long time now and know its rich history. It's a proud moment for me to take on this role at Worcestershire. Kadeer and I always work hard to improve and develop players and we look forward to working more closely with our professional squad and the famous production line of talent from our academy.
"All of our efforts will be on helping these players fulfil their potential, both individually and, most importantly, as a team.
"I feel that we've already got a good relationship with our members and supporters, and we'll be looking to build on that and look forward to sharing lots of memorable moments with them."
Kadeer added: "This is a wonderful opportunity for me and I'm really excited for the future of the club too. The members and supporters really know their cricket and have always been really welcoming to me, and I look forward to spending much more time with them going forward.
"We've got a young, hungry squad, and everyone has their best days in front of them. I'm excited about working with the players on a full-time basis."