Joshua and Fury were supposed to go head to head later this year but a court arbitration ruling means that the latter must first fight Deontay Wilder.
Fury will now clash with Wilder in Las Vegas on July 24 while Joshua will defend his IBF, WBA and WBO gold against Usyk in September.
In March, two-time heavyweight champion Fury confirmed a two-fight deal with Joshua, with the first to be hosted in Saudi Arabia this July or August.
The pair spent the next few months exchanging social media jibes, with Joshua telling Fury that his fans were waiting for him to officially sign off the fight.
“Unfortunately, his team let the whole boxing world down,” Joshua told Sky Sports.
“I will still be here, still ready to put on a show.
“End of the year. Let me get past Usyk first. But with or without Usyk in my life, I will fight Fury.
“Usyk isn’t the be-all and end-all.
“Usyk doesn’t determine the Fury fight. The Fury fight has to happen. It’s a big fight, bigger than boxing, bigger than the belts.
“It will happen. After the Usyk fight, after I defend my belts.
“The fight will be bigger, better than what it would have been.”
Fury has also refused to rule out a rearranged bout with Joshua.
He will first fight Wilder but has said that Joshua and his team will then “get their five minutes of fame”.
Last week, Fury told Sky about Joshua’s team: “I’ve got to get past Wilder then those guys will get their five minutes of fame. I’ll give them the biggest beatdown they have had in their lives.
“I’ll deal with these guys later – the most important thing to me is Wilder, the most dangerous heavyweight in the world who would knock out Joshua in the first round. As I’ve told [Joshua’s promoter] Eddie Hearn, the difference between me and them is they are businessmen and I’m a Spartan.”
Joshua responded: “I am 100 per cent sure that I will fight him and win. You’ve got to ask him the same question.
“I’m not too sure [what he would say].
“We did everything. During a global pandemic – the toughest time to organise a fight like that – we managed to have 20,000 fans available, a site fee, the media ready, my name was on the contract, I was in training.
“Then boom, they cancelled. I stay ready to fight them all because I’m a throwback fighter.”