Richard Cusworth, who is originally from the West Yorkshire city, and his Salvadoran wife Mercedes Casanellas have finally taken their true son, Moses, back to their American home, a year after his birth.
The couple spoke to the BBC on Thursday and said they still have no idea how the swap happened.
Ms Casanellas was suspicious when she noticed the features of her newborn differed from those of the boy doctors handed 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 she took a DNA test which said there was a 0 per cent 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 tracked down 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 fell to the floor involuntarily when she later got the results of the DNA test.
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?”
His wife added: “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 tests showed that it was “impossible - 0.0%” that she was the mother of the baby they had brought home, she said.
The UK ambassador to El Salvador, Bernhard Garside, told the BBC Radio 4 Today programme how he helped the Cushworths get their baby returned.
“When we first got involved in this story, it looked very, very much like an uphill struggle. I’ve been involved in consular cases before which have been less complex but have not come out the same way. My fear was that we really weren’t going to see a happy conclusion to this.
“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.”
He went on: “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 added: “From a family perspective, this has been very, very tough. This is every parent’s nightmare. The bureaucracy of the El Salvadorean system always seemed to conspire against them, but with the help of the Supreme Court judge and some good old-fashioned diplomacy we finally managed to get leverage, and we got the result we wanted.
“It is deeply rewarding and I am very, very pleased for them.”
Mr Garside said he understood the El Salvadorean authorities have now concluded their investigations into the incident and found that “there was no criminal element involved in any of this and it was simply a mistake”.