The 25-year-old put pen to paper on a two-and-a-half-year deal at Whaddon Road just minutes before the 11pm deadline.
Manager Neal Ardley told the club's official website: "Byron has been brilliant for us.
"There was interest from Cheltenham earlier in the season but Byron wanted to stay and prove people wrong by scoring goals at Wimbledon.
"He has been hit by a back injury lately but he has all the attributes to be a top striker.
"I would have had no problems with Byron staying if it never worked out with Cheltenham but he has met with them and it appears to be the right move for him."