While both clubs discussed compensation, Newcastle manager Rafa Benitez revealed he was in the dark over reports that Hearts had made his assistant coach, Cathro, their number-one choice to replace Neilson.
Former Dundee United coach Ian Cathro is being linked with the manager's position at Hearts, should it become vacant.
The former Hearts full-back gave journalists the slip when a member of his coaching staff, believed to be head of sports science John Hill, donned a Neilson face mask along with the outgoing head coach's initialled coat as he got into his vehicle and drove off in front of the cameras.
Neilson will take charge after Saturday's FA Cup second round match at Charlton.
The Dons, who parted company with Robinson on October 23, and have since had Richie Barker in caretaker charge, are 19th in the League One table.
The Paisley born boss will be remembered for nothing else but progress with the Edinburgh club and he leaves Hearts in second place in the Scottish Premiership table.
Neilson was confirmed as the new MK Dons manager on Friday evening
MK Dons chairman Pete Winkelman said: "I'm hugely excited".
"Hopefully the fans are ready for the ride. With the squad that we've got here, we should be winning football matches - our job is to help them do that".
"Ann Budge, the board of directors and everyone at Heart of Midlothian Football Club would like to sincerely thank Robbie and Stevie for the enormous part they have played in helping to re-establish the club in the upper echelons of Scottish football".
Hearts have made it clear the 30-year-old Scot is the man they want to fill the head coach's role and Cathro is keen to take the opportunity.
He guided them to the Championship title in 2014-15 - his first season as a manager - and then a third-placed finish in the top flight last term. "They worked with us to make sure it was as smooth as it could be, given that it's in the middle of the season". "It's good that Jon and Andy are going to take the team as it allows me to focus on trying to get the right person in".