Dad Abdur Rahman Rumel was driving home at midnight with his son Mohammed, five, pregnant wife Shiri and her sister when a police car suddenly boxed in his Lexus.
Police quickly realised their mistake and said the family's car registration had been cloned by the wanted armed robbers.
But Abdur said he and his family have been left traumatised by the blunder on Sunday night in Bradford, West Yorkshire.
Mohammed suffered whiplash and Shiri, who was 10 weeks pregnant, had an anxiety attack - and claims a paramedic told her to 'f**k off' when she asked for help.
Abdur, 35, said: "My son was not feeling well so my wife and her sister took him to the hospital and was waiting for four hours so I went to join them.
"Driving back I saw a car pulled over at the side of the road. Then it began to follow me. Shortly afterwards another car overtook me and they were flashing at me to pull over.
"I didn't understand what was going on; I was driving slowly because my family were in the car.
"When I stopped an officer got out of the car and grabbed my hand, pulling me out of my car. I couldn't believe what was happening. My son was crying and my wife was shaking like a leaf - she is ten weeks pregnant.
"I said 'is everything okay?' Suddenly the officer had this look on his face - he realised he had got the wrong person and told us another car had replicated our number plate and they were trying to track them down.
"He saw I was shaking, but he didn't even apologise to us. We are humans - you can only imagine how it felt."
Paramedics were called to the scene when Mrs Rahman began to have an anxiety attack caused by the incident, while Mohammed and his father suffered from whiplash.
Abdur said they challenged paramedics about the incident adding they were met with staff telling them to 'f**k off'.
Shiri, 34, said: "We were barricaded in and the car was surrounded by police, there must have been eight or nine of them.
"I couldn't see any guns, but the police told us the car that hit us was an armed response vehicle.
"We couldn't understand what was going on. I was shocked. Terrified wasn't the word for it.
"I started to have an anxiety attack. I was finding it difficult to breathe and was panicking. My son was crying in the back."
Her husband said the collision had caused 'damage to the bumper and inside' of the family's black Lexus.
He said: "It's really shaken us up. Now, when I'm driving, I keep thinking I'm being followed and pull over by mistake. It was really scary."My wife is not okay. She is traumatised and it left her crying."
West Yorkshire Police said: "At 11.59pm on Sunday, February 1, police receive information about a vehicle which was believed to have been used in an armed robbery outside of West Yorkshire.
"Following inquiries, a vehicle bearing the suspected number plate was sighted in Bradford.
"In accordance with the information received armed officers were deployed to stop the vehicle safely, which resulted in a minor collision.
"Ambulance crews were also requested to attend to a pregnant woman who was inside the vehicle. Information was later received that the suspect vehicle had been recovered."
Ben Holdaway, of Yorkshire Ambulance Service NHS Trust, said: "We are aware of the incident that took place in Bradford.
"We would like to reassure members of the public that patients' needs are at the heart of everything we do and providing a safe, responsive and high quality service to the people of Yorkshire is our main priority.
"However, any attacks on our staff, both verbal and physical, are completely unacceptable and we operate a zero tolerance policy towards violent and aggressive behaviour."
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