Sir Tom's coffin, draped in a union flag, was carried to the crematorium on Saturday by soldiers from the Yorkshire Regiment while a Second World War-era C-47 Dakota performed a flypast.
Sir Tom's 100th birthday celebrations previous year included a Spitfire flypast and he was photographed punching the air as it went past.
A version of the song Smile, recorded by singer Michael Buble, was to be played at the funeral.
Defence Secretary Ben Wallace said: "In national emergencies ordinary people do extraordinary things and inspire us all to pull together to overcome adversity".
"Few will have heard of Sir Tom before this crisis but his contribution and example now lives on in us all".
In a statement, his daughters said Captain Sir Tom had spoken "openly about his death and his funeral" over the past year and had "wondered out loud if perhaps the interest in him over the last 12 months would mean we would need to have more Victoria Sponge cakes available for the extra guests".
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.
Australia says Facebook will lift its Australian news ban
So what Microsoft is doing here is asking the European Union to adopt similar legislation throughout that jurisdiction. Australia and the tech giants have been in a stand-off over the legislation widely seen as setting a global precedent.
Soldiers performed ceremonial duties at the service for the 100-year-old, whose charity walk inspired the nation and raised nearly £33 million ($60m).
This was followed by a firing party of 14 each firing three rounds in unison before a small service got under way.
The service, at Bedford Crematorium, was attended by eight members of Sir Tom's immediate family - his two daughters Hannah Ingram-Moore and Lucy Teixeira, four grandchildren and his sons-in-law - all wearing masks.
The family set up an online book of condolences that gathered thousands of messages. The soldiers then marched out and left his close family for the service, to the sound of Moore singing "You'll Never Walk Alone" in a charity single he recorded with Michael Ball.
His family has asked the public not to attend his funeral and "continue to support the NHS by staying at home".
Captain Sir Tom Moore's daughter, Lucy, said her father would have been "chuckling" after plans were unveiled.
He had asked for his epitaph to be "I told you I was old, " inspired by comedian Spike Milligan whose gravestone says "I told you I was ill".
His ashes were later to be interred in a family plot in his native Yorkshire in northern England, with his parents and grandparents.
While two of her children had already died by the time she was found in the camp, she was heavily pregnant and soon to give birth. Shamima Begum will not be allowed to return to the United Kingdom to fight her citizenship battle, the Supreme Court has ruled.
He said "President Trump has flaws" but that he had made the party more approachable for working-class Americans. CPAC is an event organized by the American Conservative Union, whose chairman, Matt Schlapp, is close to Trump.
Caddick vanished from her Dover Heights home in Sydney's eastern suburbs on November 12 a year ago and has not been seen since. But liquidators say the account numbers on the documents either did not exist or were not associated with the actual investor.
The groups have developed numerous, often low-priced ways of creating headaches for the United States and its allies. The UN did not immediately respond to Middle East Eye's request for comment on the legality of the air strikes.
February 26, 2021: Horgan announces the project's cost has grown to $16 billion, and it won't be completed until 2025. Indian Chiefs suggested the decision violated the premier's commitment to a United Nations declaration.
37 new COVID-19 cases confirmed on Vancouver Island
County, the inflammatory syndrome has disproportionately affected Latino children , who account for 71% of the reported cases. The total case count rose to 28 409 036 infections , with 508 088 deaths as of Thursday, according to the CSSE tally.
Intelligence report on Jamal Khashoggi's murder released
Former president Donald Trump paid little attention to Riyadh's human-rights record, including its military intervention in Yemen. He also hopes to address other challenges, such as fighting Islamist extremism and advancing Arab-Israeli ties.
Attorney: Equality Act Unfair for People of Faith
The bill is one of President Joe Biden's top legislative priorities that he wants passed in his first 100 days in office. On the surface, the goal of the Equality Act (House Resolution 5) supposedly is to protect people from discrimination.
Leroy Merlin en Russie est au bord du scandale ?
La part du réseau Leroy Merlin sur le marché russe de "l'équipement ménager" approche les 20%. Beaucoup d'entre eux se plaignent des manifestations de corruption et des violations de l'éthique des affaires.
Fils de légende de la Formule 1 a dépassé son père
Nul doute qu’elle offrira à Mika la meilleure option pour poursuivre sa carrière dans le sport automobile. Et il n'est pas Andrei Arshavin, pour répondre à la question sur les attentes facilement et naturellement.