If the remains are confirmed to be those of the 21-year-old west Belfast man, it will be the first time one of the Disappeared will have been recovered in Northern Ireland.
His family held a short prayer vigil at the scene on Tuesday evening, ahead of the removal of the body which is expected to take place on Wednesday.
Mr Wilson, who had special needs, was murdered and secretly buried by republicans after going missing from his Beechmount home in August 1973.
The IRA leadership has never admitted responsibility, but searches by the Independent Commission for the Location of Victims' Remains were last week launched for the first time in Northern Ireland - triggered by a tip-off believed to have come from within the republican movement.
This is a special day for our family - Peter has been missing for 37 years. For 37 years, we have missed him and have often wondered what happened.
Hopeful that the remains are indeed those of their loved one, the Wilson family said they would be relieved to finally give him a Christian burial.
"Right now it has not completely sunk in," they said. "We were hopeful when the dig began, but we did not expect a result so soon."
In a statement, the ICLVR added: "The recovery is ongoing and the formal identification process will take some time."
Sinn Féin leader Gerry Adams said his thoughts were with the Wilson family.
"I would hope that confirmation would be speedy to ease the burden of the final wait the family will endure," he added.
"I again would repeat my appeal that anyone with any information which might help other families locate remains and find closure should bring that information forward."
Fifteen men and one woman are believed to have been murdered and "disappeared" during the Troubles - to date, eight have been found at rural sites in the Republic of Ireland.
The search for Mr Wilson at a beach in the seaside village is not only the first north of the border, but the first in a populated area.
It has also emerged that the victim's family had often visited the holiday spot - never suspecting that their loved one could be buried there.
Earlier on Tuesday, Mr Wilson's sister had carried a wreath in a silent walk at Stormont in memory of the Disappeared.
Alongside Patricia Gearon was Kathleen Armstrong, who carried a black wreath with white lilies symbolising those who have not been found - this year, one lily had been removed following the recovery of her husband Charlie's body in July.
Mrs Armstrong's friend and neighbour Mary Evans also attended the event, as she waits for DNA confirmation that remains found last month were those of her son Gerry.
"Our walk at Stormont each year is a reminder that our plight is ongoing and that every effort needs to be made to bring our loved ones home for Christian burial," said Anne Morgan, whose brother Seamus Ruddy is believed to be buried in France.
Active investigations by the ICLVR into the whereabouts of the remaining Disappeared are continuing.
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