The officers were blocked in their van with Bird's gun pointing at them after pursuing him on his murderous rampage through the town last Wednesday morning.
They managed to escape but lost sight of Bird's grey taxi.
The pair had taken over the pursuit from another unarmed officer, who heard shots and commandeered a car to give chase.
He abandoned his pursuit to give first aid to a cab driver Bird had shot as he passed him in his taxi.
Cumbria's Deputy Chief Constable Stuart Hyde insisted today none of the officers could have stopped Bird, who went on to kill a further nine people.
He said: "This incident was unprecedented and exceptional circumstances were fast-moving and highly dangerous.
"Had any officer or member of staff had the clear opportunity to stop Bird I am confident they would have taken it."
By the time the first officer at Whitehaven Police Station heard shots, divorced father-of-two Bird, 52, had already killed his twin David, solicitor Kevin Commons and his friend and colleague Darren Rewcastle.
The first police sighting of Bird came when the neighbourhood policing officer heard shots at around 10.30am.
He found Bird's grey taxi with a shotgun pointing out of the front passenger window.
He alerted colleagues and then got into the nearest vehicle, driven by a local man, to follow Bird.
He saw Bird slow down as he passed a taxi travelling in the opposite direction.
As the two cars passed, Bird shot the driver of the taxi.
The officer went to the assistance of the injured man, giving him first aid and getting him and his female passenger, who had also been shot, out of Bird's line of sight.
The two other officers in a police transit van then took over the pursuit and continued provide sightings to the control room and the armed response vehicles on route to the scene.
But Bird pulled into a driveway, turned and pointed the gun directly at the officers.
The force said in a statement: "They were forced to protect themselves after it became clear they could not reverse due to the traffic that had built up behind them.
"We believe he (Bird) then drove off at speed.
"Despite having just witnessed a shooting and having a gun pointed at them, the officers attempted to follow him and despite asking passers-by where he had driven, were unable to locate him again.
"They continued to stay in the area where further reports of Birds location were given.
"These officers were at the time unarmed and in a vehicle not suitable for a high-speed pursuit.
"Whilst they had no opportunity to bring an end to Bird's rampage they provided valuable information regarding his whereabouts and likely route.
"They also provided immediate assistance to those shot and injured in the area."
Details of the police pursuits came as inquests into the deaths of Bird's 12 victims were opened and adjourned.
David Roberts, the North and West Cumbria coroner, said initial post-mortem examinations revealed that all Bird's victims died from "wounds from firearm".
The coroner also opened and adjourned the inquest into Bird, who shot himself in woodland at Penny Hill Farm, Boot, Cumbria.
Addressing a packed courtroom 2 at Whitehaven Magistrates' Court, Mr Roberts said: "Today is obviously a very solemn occasion for the town of Whitehaven and west Cumbria generally and the surrounding villages affected by the incidents on June 2, when these deaths all occurred.
"I have received preliminary conclusions of death from the pathologist in respect of all 12 individuals as 'wounds from firearm'," he said.
In the early hours of last Wednesday, self-employed taxi driver Bird, 52, claimed his first victim.
It is thought a simmering grudge against his twin brother David over money from their late father sparked the mass killings.
Police also confirmed the taxman was investigating Bird.
After blasting David to death at his home, Bird went on to target others police believe he may have had bitter feuds with in the past.
He killed family solicitor Mr Commons and pumped shots at fellow cabbies outside the taxi rank in Whitehaven, claiming the life of Mr Rewcastle.
He then went on an hour-long rampage killing random passers-by. But details emerged on Saturday of another man who was seemingly on Bird's hit-list but who had a lucky escape.
Neighbours of Jason Carey, who went scuba diving with Bird, said the killer banged on his front door but Mr Carey, who lives in Wilton, had been on the night shift and did not answer.
After failing to rouse Mr Carey, Bird continued driving, firing his shotgun and .22 rifle from the window of his taxi.
Police launched a frantic manhunt which ended when Bird took his own life in remote woodland after abandoning his car.
The other victims were Kenneth Fishburn, 71, Susan Hughes, 57, James Jackson, 67, Jennifer Jackson, 68, Isaac Dixon, 65, Michael Pike, 64, Jane Robinson, 66, Jamie Clark, 23, and Garry Purdham, 31.
Bird's sons Graeme and Jamie said in a statement yesterday that they were "mortified" by what he had done.
Bird's elder brother Brian described the loss of both his brothers as "devastating" and said he did not know what had sparked the crimes.
Mr Hyde described the incident as an "incredibly unusual and unprecedented event".
"Certainly one of them (officers in van) tried to use the cover of the vehicle to protect themselves, to ensure they weren't shot or at least there was some protection for them but they did not have a great deal of opportunity to do that before the taxi drove off," he told reporters.
"Those officers were putting themselves in imminent danger, they could see the man was armed and he had just shot someone in front of them.
"I think that's a situation they were not used to and certainly a situation I hope they never come across again in their service.
"They are working but clearly they are very shaken by this. Anyone would be, no matter what their role, training or what job they did.
"If someone has pointed a weapon at you, of whatever description, you would inevitably be highly concerned about that."
Mr Hyde repeated his assertion that no police officer had the opportunity to end Bird's rampage any quicker.
"I think with hindsight it is possible that people might believe we could have got there faster or done other things," he said.
"The reality of it was this was an exceptional incident in exceptional circumstances.
"These are ordinary neighbourhood officers who go about their normal business protecting the people of Whitehaven.
"They did everything they could to get to the scene.
"Armed officers were deployed within two minutes of the first call, going from the police station, which is just round the corner (from Duke Street), possibly within seconds, is a pretty fast response in anybody's world."
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