Health officials said the rescued boys would now spend at least a week in hospital and about 30 days recovering at home after their ordeal.
Mr Stanton, aged in his late 50s, has dismissed suggestions that he and the team are heroes and should be honoured.
Former Navy seal Chaiyananta Peeranarong told AFP news agency: "Some of them were asleep, and some of them were wiggling their fingers, kind of groggy - but they were breathing".
"How many of you?" he asked, a torch beam scanning the boys crowded on the bank.
Isn't that wonderful? The Thai rescue mission has taught us one thing. They got far enough inside that rising floodwaters, not unusual at the cave this time of year, blocked their exit.
Previously, Thai authorities had kept details of the incident including how a group of boys, many of whom could not swim and had no diving experience, could navigate the treacherous narrow and submerged passageways of the Tham Luang complex, even with expert diving support.
Arriving at Heathrow Airport on Thursday, Mr Volanthen said it was a "relief" but played down his heroics.
Pushing a cart stacked with his luggage, Volanthen paused to speak with reporters about the rescue mission.
"We just needed them to know how to breathe and not panic in the water", he said.
"What we do is very calculating".
Rescuers are sent inside Tham Luang Nang Non cave to continue the rescue after the 12 boys and their soccer coach have been found alive in the cave where they've been missing for over a week after monsoon rains blocked the main entrance on July 03, 2018 in Chiang Rai, Thailand.
"We were very, very pleased it worked out quite so well".
On Tuesday evening, all 12 boys and Chanthawong had been rescued following a three-day operation by the Thai Royal Navy and Thai Navy SEALs.
"When the divers came out we were obviously congratulating them and helping them with their gear, but I think they just wanted to get out of there because they had been in there a long time".
The operation claimed the life of Thai navy diver Saman Kunan, who died while replenishing oxygen canisters. The team became captured within a cave in Thailand on June 23 and survived for 18 long days before being miraculously rescued by a courageous group of global divers.
Each boy was strapped to one of two rescue divers tasked with shepherding him, and bundled onto stretchers to be carried through the dry parts. "It will be good to get closure".