The funeral opened with a rendition of You’ll Never Walk Alone that Sir Tom recorded with Michael Ball and the NHS Voices of Care Choir that reached number one last year.
His daughter Hannah Ingram-Moore talked about how his surprise fame saw him “appear on the front of every paper, and on channel around the world”.
She added: “We could not have asked for a magical childhood. We felt so loved and so safe.”
His daughter Lucy Teixeira paid tribute to him, saying : “You have always influenced me with your strength, your energy, your drive, to get out of bed with a spring in your step and a purpose in mind.
“I know you will be watching us chuckling, saying ‘don’t be too sad as something has to get you in the end’.
“Daddy, I am so proud of you, what you achieved your whole life and especially in the last year, you may be gone, but your message and your spirit lives on.”
His grandson Benjie said he had taught him “the power of positivity and kindness”.
He said: “Growing up with you every day, the smell of porridge as I came downstairs was almost a sign to know everything was okay.”
Benjie added his grandfather sneaking leftovers to the dogs “filled the morning with more than a bit of joy”.
The service also included a poem the veteran had chosen asking that “no mournful word be said”.
It went on: “I was happy all my life. My two daughters I married right and their children I rocked at night. Death nor sorrow never brought cause for one unhappy thought.
“Now and with no need for tears here they leave me full of years. Leave me to my quiet rest in the region of the blessed.”
The celebrant conducting the funeral said it was “incredible to think” that people from “almost the whole world” contributed to his fundraising efforts.
They said: “As wonderful as we think our NHS is, people from other countries really aren’t going to be interested in our health, so it seems obvious to me that they were really investing in Captain Tom and the values he stood for.
“He was a proud British veteran and a gentleman, he lived in a multi-generational environment, not only would that have kept him young, but also symbolises the importance of family to him.
“What sacrifices did he and his peers make in defence of our freedom, a man with a strong moral compass, a strong work ethic, a sense of pride and an indomitable spirit.
“He serves as an inspiration to us all to never give up and always stay strong knowing tomorrow will be a better day.”
Earlier, members of the Yorkshire Regiment lifted his coffin from the hearse to bear it into Bedfordshire Crematorium.
A Dakota performed a fly-pass ahead of a three-round gun salute by a firing party before the coffin was carried into the building.
A number of special items were placed on the coffin, including a replica of his service cap from the Second World War and a wreath from the Yorkshire Regiment.
Also among the items are his campaign medals, including the Burma Star, and his knighthood medal stitched on to a cushion.
There is also a specially commissioned sword engraved with the motto of the Yorkshire Regiment on one side – “Fortune favours the brave”.
Engraved on the other side is his own personal motto, “Tomorrow will be a good day”.
Flags across the country were flown at half mast this morning as the nation prepared to say goodbye to Sir Tom and crowds lined the route of the funeral procession.
A version of the song Smile recorded especially for the occasion by Michael Buble will be played at the service.
The White Cliffs Of Dover by Dame Vera Lynn and I Vow To Thee My Country by Alfie Boe will also be played.
Once lockdown is over Sir Tom’s ashes will be interred at a family plot in Keighley, Yorkshire.
Earlier on Saturday the deputy Lord Lieutenant of West Yorkshire, David Pearson, laid a wreath in Keighley – Sir Tom’s birthplace – on behalf of the Queen.
The wreath featured 200 white roses around a picture of Sir Tom.
Wreaths of poppies were also laid by local MP Robbie Moore and the town’s mayor and mayoress, Peter Corkindale and Clare Abberton.
Sir Tom was made an honorary freeman of Keighley last summer when he said “it really is great to be back” as he watched a plaque unveiled in his honour.
Sir Tom, who raised more than £32 million for the NHS with a sponsored walk of his garden in the first lockdown, served with the Duke of Wellington’s Regiment during the Second World War.
The regiment later merged with two others from Yorkshire, becoming the Yorkshire Regiment, and Sir Tom was made an honorary colonel last August.
His funeral was attended by eight members of his immediate family – his two daughters, four grandchildren and his sons-in-law.
Ms Teixeira, 52, said the service would be “quite spectacular”, adding: “There’s just going to be the eight of us under full Covid restrictions, we will honour him the best way we possibly can.”
There are plans to plant trees around the world in his honour, with Ms Teixeira hoping the Trees for Tom initiative will result in a wood in his home county of Yorkshire and the reforestation of part of India, where he served during the Second World War.
“My sister and I have been creating the funeral that my father wanted,” she said.
“He was very clear in his wishes and if he could have been put into a cardboard box, he would have done that, rather than chop down a tree.”
The family has urged people support the NHS by staying at home, with many thousands of people signing an online book of condolence instead.
Once Covid-19 restrictions permit, they will inter Sir Tom’s ashes in Yorkshire, with his parents and grandparents in the Moore family plot.