The WBA, WBO and IBF heavyweight champion furthered his reputation with Saturday's clinical seventh-round stoppage of Russian Alexander Povetkin at Wembley, where he is already scheduled to fight again on April 13.
Joshua has repeatedly said that Wilder would be his chosen opponent, even with the WBC champion scheduled to fight Tyson Fury in America on December 1.
Previous negotiations for a lucrative fight between them for all four titles have ended in frustration, and Joshua's domestic rival Dillian Whyte is on standby should they do so again, contributing to Hearn's demand that they are soon finalised.
— Anthony Joshua (@anthonyfjoshua) September 22, 2018
Should Fury defeat Wilder, then the American's rematch clause with Fury would almost certainly be activated and regardless rule both fighters out.
The victory they expect for Wilder would also give Fury little immediate appeal, so Hearn said: "Being British, we'd like Fury to win, but for April, Wilder must win if that's going to happen.
"We're not willing to wait until December to see. A deal must be done in advance of that, subject to him winning. The offer that was made to us (previously, by the Wilder camp) will look like absolute peanuts when this fight is made.
"That is the biggest fight in boxing; Wilder-Fury is a really good fight to see who's the second best heavyweight in the world. Joshua-Wladimir Klitschko was the biggest fight in British boxing history, and Joshua-Wilder would eclipse that. We want that now.
"We're not waiting for timewasters; we understand if they lose we'll have to find another opponent, but if you win, we're not waiting until December. These aren't negotiations that will take 24 hours, and Joshua's career's not being slowed down. If they don't want to do that, we'll fight someone else.
"Tyson Fury is the least entertaining fighter I've ever seen. He's never been in a good fight, apart from against Steve Cunningham (in 2013), when he got knocked down."
The 28-year-old Joshua confirmed he had not broken his nose despite it bleeding heavily after Povetkin hurt him with a big left hand in the opening round.
Fury has already said on social media that Joshua will never fight him, and the champion added of Wilder: "I don't want to wait until December. For him to finish his fight, have his rest and then start negotiating.
"I'll start training again in early January, so I want to get it pencilled in as soon as possible, and those negotiations take a long time. If Wilder's not serious, there's other people out there; when he's ready, we're ready.
"Good luck to them both (Wilder and Fury) – boxing needs it. I've had the burden of the heavyweight division on my back for some years, because it was all about me fighting Wilder, Fury, Klitschko, Dillian, Povetkin. That's all they were interested in – me fighting them all.
"So I'm happy those two are fighting. April 13 is booked, so whichever heavyweight is serious, we can look at making a deal. I have no interest in who wins; I'm not fussed."