Their lively rendition of Take That's Rule The World drew warm reaction from the judges.
Panellist David Walliams said he hoped the singers from the Fleetwood school would win the show.
"It was joyous, it was magical," he said.
Amanda Holden said she was moved to tears while Simon Cowell said they were the most fun act to have made a Britain's Got Talent final.
But Mr Thackery, 89, earned a standing ovation for his moving rendition of Love Changes Everything.
And said he was "astonished" to have won, beating magician X to win the ITV show.
"I don't know what to say," he told hosts Ant McPartlin and Declan Donnelly.
"I'm stuck for words, for a change."
X said it felt "amazing" to be runner-up.
He said: "To come second to someone like Colin, who has given his life for us ... I bow to him.
"Deepest respect. I absolutely love the man. He deserves to win."
Holden said she knew Thackery, 89, was not taking part because of the fame and glamour.
"I know that this is because you served your country and I hope tonight that they serve you," she said.
Asked what it would mean for him to sing at the Royal Variety Performance, Thackery said: "I served my Queen for 25 years and to think I could sing for her would be the end."I could die happy."
Eleven acts took to the stage this evening in the hope of winning £250,000 and a spot on the bill of the Royal Variety Performance 2019.
PC Dave Wardell and his retired police dog Finn opened the Britain's Got Talent final by announcing the Queen has signed a law to give service animals greater protection from attacks.
Finn, who is now retired, saved Mr Wardell's life when a robbery suspect they were pursuing turned on them with a knife in 2016. The dog suffered serious stab wounds to the chest and head but did not let go until reinforcements arrived.
The duo have made it through to the final of the ITV show with a series of performances based on the idea that Finn can read minds, and were the first act to take to the stage on Sunday night.
After finishing their magic act, Mr Wardell told the audience: "After our first audition aired the Queen stayed up late one night and she signed off on Finn's Law.
"So on June 8, Saturday next week, Finn's Law becomes a law."
Cowell said the pair were one of his favourite acts, adding: "I love you and I am absolutely in love with Finn.
"He is adorable."
Walliams added: "With the law being passed you are winners already."