The experienced front-rower has enjoyed a strong season in the Championship with the Blues, helping them to reach the play-off final where they will face Newcastle. He also featured for Saracens in the LV= Cup.
Reeves, a lance corporal who is part of the 33 Engineer Regiment bomb disposal unit and has served with the British Army in Afghanistan, will be given a sabbatical period by the Army in order to play for Wasps.
Wasps rugby director Dai Young said: "Ricky is an experienced player who has been a key part of Bedford's excellent season this year and is very keen to be given a chance in the Premiership.
"He will be a very useful addition to our squad and we are looking forward to seeing what he can do at the top level."
- Sports & Recreation