A senior officer has apologised to Andrew McQuillan, son of Alan McQuillan, and said there had been a high number of incidents on Tuesday evening and no crews had been available.
Mr McQuillan said he contacted police just after 6pm.
Officers did not respond until after 9am yesterday morning.
Mr McQuillan took to social media, posting on X that he had reported a male threatening people with a firearm to police on Tuesday evening at 6pm, and officer had not attended 14 hours late “citing lack of officers”.
He queried whether this was acceptable.
He later spoke to BBC Radio Ulster's Stephen Nolan show about his experience.
Mr McQuillan said he had been in his office at about 6pm on Tuesday evening when he received a call from someone he did not know threatening him.
“Initially, they sent me a WhatsApp message saying they were going to chop me up and my family, they then started phoning constantly from a withheld number, I answered and they basically said they had a gun, they were on their way to shoot me and they were on their way now,” he said.
“This was witnessed by other people because I put it on speaker phone. They called me a derogatory name for members of the Catholic community as well.
“Myself and the people with me were fearful.”
He said he called 999 and was told police were en route.
“The initial 999 call was great, they told us to lock the doors and stay inside, and police were on their way. Twenty minutes later, nothing,” he said.
“I phoned police back and they said they were incredibly busy, they said, 'we don't have as many police as we used to so we can't come to you just yet'.”
He also said he was told that an inspector had downgraded his call from an immediate response call to an emergency call.
He added that he wanted to speak out about the incident because he was "disgusted".
District commander for Lisburn and Castlereagh, Superintendent Kelly Moore, said she had apologised directly to Mr McQuillan.
“Police received a report at 6.17pm on Tuesday August 22 of threats against a man,” she said.
“Due to the high volume and nature of ongoing incidents at that time within the Lisburn and Castlereagh area, unfortunately, no crews were available to attend.
“I have spoken to the reporting person directly and have apologised for officers not attending at the time and understand the distress this has caused him.
“At 9am this morning, Wednesday August 23, officers attended the address to speak with the reporting person and to offer crime prevention advice.
“We have launched an investigation and we are treating this as a sectarian hate crime.
“As part of our policing response we will be increasing patrols of the area to provide reassurance.”