Soldier F will face charges for the murders of James Wray and William McKinney and the attempted murders of Joseph Friel, Michael Quinn, Joe Mahon and Patrick O'Donnell, the Public Prosecution Service (PPS) said.
However, it declared that 16 other former soldiers, and two suspected ex-members of the Official IRA, all of whom were also investigated, will not face prosecution.
Sinn Fein and the SDLP said they shared the families' disappointment at only one prosecution.
However, the DUP and TUV leader Jim Allister said there was too much focus on security force killings and that IRA victims were being forgotten. Mr Allister asked: "Why is Bloody Sunday more important than Bloody Friday?"
The Bloody Sunday families gathered in the Guildhall yesterday to hear the PPS decision.
Ms Nash said: "I left home certain that I would get justice for Willie. I am shocked, absolutely gutted, that I did not.
"I am delighted for the Wray and McKinney families, but the decision is a huge slap in the face for the rest of us. But we have no intention of giving up. We will ask the PPS to review the decision not to prosecute the others, and we also plan to mount a legal challenge in the courts."
John Kelly, whose 17-year-old brother Michael was killed, said: "It is unbelievable that only soldier F will be prosecuted. It is scandalous. In my view every one of them should be prosecuted for murder and attempted murder.
"We had heard rumours that there would be four - and we were disappointed at the thought of that - but I am totally devastated in the fact that Soldier F has got away with Michael's murder.
"In saying that, he will be prosecuted for two other murders and their victory is our victory and I will take solace from that."
Michael McKinney, whose brother William is one of the two murders Soldier F will be charged with, said he shared the disappointment of the other Bloody Sunday relatives, while they shared his family's relief.
"I think the soldier will be charged within the next few days and things will be set in place, this has to start happening right away," he said.
There was a mixed reaction among unionist politicians. DUP MP Gregory Campbell said the IRA's murder of two RUC officers in the days leading up to Bloody Sunday was being airbrushed from history.
"The facts also remain that 90% of deaths in Northern Ireland were at the hands of illegal terrorist groups who existed solely to murder and cause destruction," he said.
"There is still a disproportionate focus however on the small proportion of the 10% of deaths attributed to those who were attempting to serve the community in difficult and often very dangerous situations."
Mr Allister said: "While insatiable demands for wholesale prosecutions may have perished on the rock that prosecutions are only possible where there is sufficient evidence, the hierarchy that has elevated Bloody Sunday families above all others is hard to take.
"In 1972, we also had Bloody Friday, but IRA murders don't count it seems when it comes to this distorted dealing with the past. The pursuit of soldiers while terrorists continue to go scot-free is now very much part of the rewrite of history so promoted by IRA/Sinn Fein, who themselves still withhold information on multiple murders."
Ulster Unionist MLA Doug Beattie said the decision to prosecute must be respected and nobody was above the law.
"The PSNI looked at the evidence, passed it to the PPS and they decided - based on the evidence - to prosecute one former soldier. This is the law, it must be respected by all and it is now for the courts to decide," he said.
"Nobody is above the law and nobody should be held to a greater or lesser standard than anyone else. My thoughts remain with the families of all the innocent victims of the Troubles."
Sinn Fein deputy leader Michelle O'Neill said that her party shared the families' disappointment and "incredulity" that only one soldier would be charged but that prosecution was still "a significant achievement".
SDLP leader Colum Eastwood said the decision to prosecute only one paratrooper did not vindicate the actions of the others.
The families' justice campaign had been met with "prevarication, equivocation and obstruction at every level", he added.
Kenny Donaldson of Innocent Victims United said: "Our organisation holds a consistent policy on criminal violence whether perpetrated by terrorists or individual members of the security forces.
"The PPS has found that tests have been met for Soldier F to face prosecution for two deaths and a number of other attempted killings; the law must now run its course".
But he added that the "overwhelming majority" of security force members had "performed their role with honour, integrity and immense restraint".
Lowry Mathers, whose wife Joanne was shot dead by the IRA in Derry while collecting census forms in 1981, said: "It's all very well getting justice for Bloody Sunday. But what about me? What about my son?
"There is absolutely no justice for us and that leaves a bad taste in my mouth," he said.
Alliance leader Naomi Long said it was a day of "very mixed emotions for the families who lost loved ones on Bloody Sunday, and for all of those injured or bereaved in the Troubles".