The mortar was found on a wall near houses in Church View, Strabane, at 8.30 am on Saturday.
Police said the bomb was “a deadly device” and believe dissident republicans were behind the “callous attempt to kill or maim” PSNI officers.
Aileen Mullan said her teenage daughter has not returned home since it was found and her son will not play outside the front of the house.
She said: “I thought what could have happened – it was right under my daughter’s bedroom window.
“My daughter is 14 and my son is seven, my son could have easily walked out the front door and lifted it, quite easily went out and lifted it and he would have been gone.”
She said when her son heard there was a bomb he “was screaming the house down.”
He told his mum he had never “seen a real bomb before.”
Ms Mullan said: “I thought he would never have to. He won’t go around the front of the house in case there is a bomb.”
She said her teenage daughter has stayed at a friend’s house since Saturday.
“That’s not what I want for my children; my 14-year-old won’t come home. She is terrified,” she said.
Ms Mullan said whoever was responsible should “come and tell me why they put my children’s lives in danger”.Detective Inspector Andrew Hamlin said the device had been an attempt to target police officers, but that it had “the capacity to kill or seriously injure anyone in the vicinity”.
“This is not the first time a deadly device has been left in a public space recently and serves to remind us all how little the terrorists responsible care for the lives of local people,” he added.
A 33-year-old man has been arrested under section 41 of the Terrorist Act over the mortar bomb find.
Local politicians have condemned those who left the mortar bomb in the border town.
SDLP MLA for West Tyrone Daniel McCrossan said the device was left on a wall close to the local police station.
Mr McCrossan described it as a “reckless, cowardly and selfish” act.
“This device obviously was very sophisticated and was placed there deliberately, not only to cause a huge inconvenience but, ultimately has endangered human life,” he said.
“This could have been much, much worse.
“There’s no mandate for it, no-one wants it.
“We want them off our streets, we want them to leave and go away.
“That’s the message that this community will be sending very strongly to those responsible.”
Mayor of Derry City and Strabane District Council, Councillor Michaela Boyle also hit out those who planted the device.
She said: “Those responsible for this serious type of activity in a residential area where there are pensioners, people with disability’s and those needing care.
“They need to wise up and move on.
“It is communities that suffer as a result.
“Thank god no one has lost their life.”