A video posted on Miss Wilcox's Twitter page shows the dark interior of the cab, where the pair can be heard saying: "We have to pay 60p?
"Why is he being so serious over 60p?"
The driver asked why the young women were "making fun of him" and demanded they wake up Miss Heath's parents so they could make up the outstanding bill.
When they refused explaining they had work in the morning, he rang his manager who then asked them whether they had anything worth 60p to offer the driver.
The women continued laughing as the driver told them he was going to take them to the nearest police station - not taking him seriously.
They wrote "I can't cope" with crying face emojis and added: "It's 60p, you're not going to die from 60p."
But when he drove off saying "For me it's not 60 f****** p" they realised he meant every word.
Pulling up to the police station with a group of patrol cars parked outside the women claim the driver kept flashing his lights.
Miss Heath told MailOnline: "He drove past police cars and kept flashing them with his lights but no one cared about the situation.
"He kept telling us he was going to die because of 60p and was quite upset about it.
"The police weren't bothered so he told us to get out of his taxi about a 20-minute walk from my house and drove off."
Intercity Private Hire has been contacted for comment.
