Assombalonga joins on a five-year deal for an undisclosed fee from Peterborough after a prolific campaign with the Cambridgeshire side last season when he scored 33 goals in all competitions.
Forest are thought to have tabled a £5million bid, rising to £8m based on appearances, for the 21-year-old striker from Zaire after he handed in a transfer request at London Road.
Earlier on Wednesday, the Sky Bet Championship side completed the transfer of Michail Antonio from Sheffield Wednesday.
The 24-year-old, who scored 13 goals in 69 appearances for the Owls, has moved to the City Ground on a three-year deal.
"I'm really pleased to be here," Antonio told Forest Player. "Nottingham Forest is a massive club, it has great potential to reach the Premier League and, on a personal level, I'm here to better myself as a player and get as far as I can.
"Forest have an amazing history and they have been pushing for promotion to the Premier League for the last four or five years. Hopefully now we can get there.
"The fans can expect lots of pace, power and goals from me this season. I'm looking to get my head down, stay fit and show the Forest fans what I can do.
"I'd love to be involved on Saturday. I've heard the game is almost sold out so hopefully, if the manager picks me, I can play a part and we can start the season off in the best way."
Antonio became a key player for Sheffield Wednesday and his performances alerted the interest of a number of other clubs, some of which were understood to be in the Premier League.
The double deal will come as welcome news to Forest manager Stuart Pearce, who was involved in a dispute with chairman Al Hasawi earlier this week after key players Karl Darlow and Jamaal Lascelles were sold to Newcastle.