A joint statement from Cardinal Sean Brady, the current head of the Catholic Church in Ireland, and Derry bishop Seamus Hegarty, said it was "shocking" that a priest could have been involved in the bombing.
"This case should have been properly investigated and resolved during Father Chesney's lifetime," they said.
But they rejected any conspiracy, saying: "The Catholic church did not engage in a cover-up of this matter."
They noted that Chesney, who died in 1980 aged 46, denied any involvement when asked about the allegations by a senior churchman.
No-one was ever charged with the murders but Chesney has long been suspected as the member of the Irish Republican Army (IRA) who masterminded the plot, although the IRA denied responsibility.
The investigation by Al Hutchinson, Northern Ireland's police ombudsman, said there was "extensive police intelligence" linking the priest to the IRA and the Claudy attack, and said many police officers wanted to pursue him.
William Whitelaw, the then British minister for Northern Ireland, and Cardinal William Conway, the head of the Catholic Church in Ireland, discussed whether Chesney could be transferred out of the province, a suggestion passed on to police.
Chesney was moved to Ireland in late 1973.
Although Hutchinson said there was no suggestion police could have prevented the bombings, he said he deplored the cover-up.
|Hutchinson said there was no suggestion police could have prevented the bombings
"The decision failed those who were murdered, injured and bereaved in the bombing," he said.
Acknowledging the reasoning for moving the priest, Hutchinson said: "I accept that 1972 was one of the worst years of the Troubles [the name given to the three decades of violence in the province] and that the arrest of a priest might well have aggravated the security situation.
"Equally, I consider that the police failure to investigate someone they suspected of involvement in acts of terrorism could, in itself, have had serious consequences."
Hutchinson described the cover-up as "collusion" but said he found no evidence of criminal intent by the church or the government, and said all those involved on the police side who might face investigation were dead.
Owen Paterson, the current British minister for Northern Ireland, said he was "profoundly sorry" that Chesney was not properly investigated.
The violence in Northern Ireland was largely ended by a 1998 peace deal but sporadic unrest still flares in the province.