Bogdanovic, 27, is averaging a career-high 14.2 points per game in his third season in the league. He's making 36 percent of his 3-pointers. The 6-foot-8 Croatian will be a restricted free agent after the season. He was originally a second-round pick of the Miami Heat in 2011 and signed with the Nets in 2014.
"Bojan is a very good shooter and a talented overall scorer whose versatility gives us an added dimension as we gear up for the stretch run,'' Wizards general manager Ernie Grunfeld said in a statement. "He is a proven starter that will provide us an added boost off the bench and allow us to be creative with our lineups."
Thornton, 29, averaged 6.6 points this season for the Wizards. He has scored 11.9 points per game over his career.
Washington (34-21) leads Atlanta by 2½ games in the Southeast Division and is third in the Eastern Conference, behind Cleveland and Boston.
The Wizards have won 18 of their past 21 games and surged up the standings after a slow start under new coach Scott Brooks.
"I like where we are as a team, but we're not satisfied," Brooks said. "We're not just going to be happy with what we've done in the first half. We want to come out and play good basketball throughout the season."
It's the second consecutive season the Wizards traded their first-round pick at the deadline; they also acquired power forward Markieff Morris from the Phoenix Suns in 2016. Morris is a starter, along with All-Star John Wall, Bradley Beal, Otto Porter and Marcin Gortat, and Bogdanovic is expected to be the sixth man.
Bogdanovic was the leading scorer in the 2016 Summer Olympics, averaging 25.3 points to power Croatia to a surprising berth in the quarterfinals as the top seed from its group. He led the Nets with 99 3-pointers and was second on the team in scoring.
"Shooting is such a big part of the game, even more so the last few years," Brooks said. "The 3-point ball is powerful, and when we win that game, we're hard to beat, like a lot of teams are."
Beal, who calls Bogdanovic "Bo," said he's excited about the acquisition.
"Strong guy, can post up, he can shoot 3s, can put the ball on the floor and can guard too, and is pretty good at rebounding and backdoor cutting," Beal said after practice on Wednesday. "I love his aggressiveness. He plays with confidence."
The deal gives the Nets a second pick in the first round, though they will lose their shot at what could be the No. 1 overall selection, since they gave the Boston Celtics the right to swap picks this year in the 2013 deal for Paul Pierce and Kevin Garnett. The Nets will get the Celtics' pick.
McCullough, 22, has spent most of the season in the NBA Development League. The 6-foot-9 forward was the Nets' first-round pick in 2015 but was delayed in starting his career as he recovered from knee surgery in his lone season at Syracuse.
Grunfeld called McCullough an "intriguing prospect" the coaching staff will continue to develop.
The trade was first reported by The Vertical.
"We want to keep charging ahead and keep climbing," Brooks said. "To win a championship, it's going to be the hardest thing that any team has to go through to win it, but it's not something you focus on day to day."
The Associated Press contributed to this report.