His pleas have been entered on the basis of diminished responsibility, his barrister Peter Joyce KC told the court.
University of Nottingham students Mr Webber and Ms O’Malley-Kumar had been returning from a night out in the city centre when they were fatally stabbed in Ilkeston Road shortly after 4am.
A short while later, Mr Coates was found dead in Magdala Road with knife injuries, while his van was allegedly stolen and used to drive at three pedestrians.
Wayne Birkett, Marcin Gawronski and Sharon Miller all suffered injuries after being hit in Milton Street and Upper Parliament Street.
The prosecution will now review medical evidence before considering whether to accept the pleas.
The family of Mr Webber, a history student and talented cricketer from Taunton in Somerset, previously described their “complete devastation”, saying he was a “beautiful, brilliant, bright young man, with everything in life to look forward to”.
Ms O’Malley-Kumar’s family said she was a “truly wonderful and beautiful young lady”, and that she would be “so dearly missed”. At the time of her death, she had been studying medicine and had previously played for England Hockey.
Their families have since pledged to create foundations in their memory to ensure they “leave a legacy”, with fundraisers raising more than £80,000 combined.
Mr Coates had been working at the Huntingdon Academy and had been months away from retiring when he was attacked and killed. His two sons said the death of the “die-hard” Nottingham Forest fan had “rocked everyone’s world”.
Following the killings, thousands attended vigils at the University of Nottingham and in Old Market Square to remember the victims.
The Independent Office for Police Conduct (IOPC) is investigating Nottinghamshire Police’s involvement in the incident, after the force referred itself to the watchdog the day after the attacks.
The IOPC previously said that an officer driving a single-crewed vehicle was responding to a related call when it came across the van allegedly being driven by Calocane.
Dashcam footage suggests the officer had sight of it for less than a minute before it hit pedestrians.
The officer immediately stopped to give first aid to those who had been hit, with the investigation considering whether the actions of the van driver were influenced by the presence of the police car.
Judge Mr Justice Turner adjourned the case until 16 January 2024.