Richard Cushworth, from Bradford, West Yorkshire, and his Salvadoran wife Mercedes Casanellas tracked down their son Moses after realising the baby they had been taking care of was not theirs.
However, the couple told the BBC they still have no idea how the swap happened.
Ms Casanellas was suspicious when she noticed the newborn’s features differed from those of the boy doctors gave her the day after she gave birth by emergency Caesarean in May 2015. She thought the second baby’s skin was darker.
After returning to their home in Dallas, Texas, Ms Casanellas took a DNA test which said there was a 0% chance she could be the mother of the baby she had been given.
The couple feared their child could have been sold to human traffickers and returned to El Salvador to make an appeal on local TV to find their son.
Their son was found after authorities ordered other new mothers have their babies’ DNA tested.
Describing the birth, Ms Casanellas said she saw her son only briefly after he was born.
“He was just passed by me and I gave him a kiss and then he was taken to the nursery and that was the last time I saw him,” she said.
The next day, nurses brought her a baby and insisted it was hers, despite her immediate doubts.
“My first impression was ‘this is not the same baby I saw last night’,” she said.
She said she fell to the floor involuntarily when she got the DNA test results.
Describing how she felt at that moment, she said: “The pain, the thought that the baby I had been nursing, taking care of, loving, that he was not mine.
“And then I had another thought, which was ‘Where is my baby?’”
Mr Cushworth told the BBC: “I just accepted it as my child. Now I look back at the pictures around the time we came to Dallas when he was three months old, and I’m shocked that I never suspected, because you can see that it’s just obviously not my child if you look at some of the pictures.
“I don’t know how I didn’t ask myself. You just don’t think about these things. Who thinks about these things?
“I think we were in love with the baby. Even when I did the DNA tests, I thought I was betraying him. That was the feeling I had — I’m betraying my son but I cannot live with this.”
The UK ambassador to El Salvador, Bernhard Garside, told the BBC how he helped the Cushworths get their baby returned.
“Once we established that this had been… an elementary swap of children and were able to quickly get the biological child back to both parents, that was a big sigh of relief for me and the team,” said Mr Garside.
“The way things are done here is that babies have their feet printed — a bit like fingerprinting —when they are born, and that was a key piece of evidence for both of the babies’ identities that were involved in this case.
“Unfortunately, neither of those footprints taken at the time of the birth were conclusive — the fact that the police could not conclusively identify either of the babies and say they were the right babies but they had been swapped.
“I asked the judge in this case to consider the evidence given through DNA, which she did take into account and it played a crucial part.”
Mr Garside said the authorities have concluded their investigations and found “there was no criminal element involved in any of this and it was simply a mistake”.
British Ambassador to El Salvador, Bernhard Garside, delighted Cushworth family reunited after baby swap pic.twitter.com/6KPOAwnH1z— Foreign, Commonwealth & Development Office (@FCDOGovUK) June 2, 2016