The Queen also spoke about personal joy, describing how she and the Duke of Edinburgh were "delighted" to welcome an eighth great-grandchild to their family - Harry and Meghan's son Archie Harrison Mountbatten-Windsor - born 200 years after Queen Victoria.
As the Queen spoke viewers saw a photograph, released a few days after Archie's birth on May 6, which captured the moment she and the Duke of Edinburgh saw Archie for the first time as his proud parents and grandmother Doria Ragland looked on.
Prince Philip spent four nights being treated before he was discharged on Christmas Eve morning, in time to join the royal family celebrations at Sandringham.
In her message to the country and Commonwealth the Queen mentioned the carol It Came Upon The Midnight Clear - performed at the end of the broadcast.
She added: "Like many timeless carols, it speaks not just of the coming of Jesus Christ into a divided world, many years ago, but also of the relevance, even today, of the angels' message of peace and goodwill.
"It's a timely reminder of what positive things can be achieved when people set aside past differences and come together in the spirit of friendship and reconciliation.
"And, as we all look forward to the start of a new decade, it's worth remembering that it is often the small steps, not the giant leaps, that bring about the most lasting change."
A string of milestones marked during 2019 were mentioned by the Queen, from the 75th anniversary of the D-Day landings to the 50th anniversary of the first moon landing.
She began her address by highlighting its theme of "small steps" as she spoke about the Apollo 11 mission.
"As those historic pictures were beamed back to Earth, millions of us sat transfixed to our television screens, as we watched Neil Armstrong taking a small step for man and a giant leap for mankind - and, indeed, for womankind," said the Queen.
She added: "It's a reminder for us all that giant leaps often start with small steps."
The Queen referenced a 'bumpy' year in her annual message
BBC
Nearby was a picture of the Queen meeting Armstrong and his Apollo 11 crewmates Michael Collins and Buzz Aldrin.
Speaking about D-Day, the Queen mentioned the "look of concern on my father's face" as King George VI knew about the secret plans but could not "share that burden".
Seventy-five years on from the historic landings, she said former enemies had met in friendship at commemorations marking the anniversary in Britain and on the continent.
The Queen added: "Such reconciliation seldom happens overnight. It takes patience and time to rebuild trust, and progress often comes through small steps.
"Since the end of the Second World War, many charities, groups and organisations have worked to promote peace and unity around the world, bringing together those who have been on opposing sides.
"By being willing to put past differences behind us and move forward together, we honour the freedom and democracy once won for us at so great a cost."
During the broadcast footage was shown of America's moon landings, national D-Day commemorations in Portsmouth, and world leaders meeting the Queen during the recent Nato reception at Buckingham Place, which marked the alliance's 70th anniversary.
Royal family Christmas service - In pictures
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The Prince of Wales and Duchess of Cornwall were featured at a French D-Day service in Bayeux, Normandy, and the Duke and Duchess of Cambridge were shown celebrating St Patrick's Day with the Irish Guards.
Speaking about how Jesus showed the world that "small steps taken in faith and in hope can overcome long-held differences", the Queen added: "Many of us already try to follow in his footsteps.
"The path, of course, is not always smooth, and may at times this year have felt quite bumpy, but small steps can make a world of difference."
The Queen's comment about experiencing a "bumpy" year is thought to be her first public reference to the personal events involving her family during the past 12 months.
During 2019, the Duke of Edinburgh was involved in a dramatic car accident, the Duke and Duchess of Sussex spoke about their struggles living in the public eye, and the Duke of York gave a disastrous television interview about his friendship with a convicted sex offender.
The Queen also paid tribute to the men and women of the emergency services and the armed forces, and remembered those on duty this Christmas working to keep the nation "safe and secure".
At the end of the broadcast footage was shown of the Queen, Charles, William and Prince George preparing Christmas puddings earlier this month at Buckingham Palace for the Royal British Legion's Together at Christmas initiative.
