A late night for George, the Pyjama Prince, as he meets the Obamas

Prince George met Barack Obama in his pyjamas as his mother and father hosted the US president and the first lady at Kensington Palace.

The two-year-old, wrapped up in check pyjamas and a white dressing gown, was allowed to stay up late and managed to show the couple his rocking horse and the stuffed toy they had given him when Princess Charlotte was born before bedtime.

It capped the end of a long day for Mr Obama and his wife Michelle, who earlier in the day had lunched with the Queen and the Duke of Edinburgh.

Mr Obama paid a glowing tribute to the Queen just a day after her 90th birthday, describing her as an "astonishing person and a real jewel".

As the Obamas arrived, to be greeted by the Duke and Duchess of Cambridge and Prince Harry in the courtyard of 1A Kensington Palace, the president shouted "oh there you are - hello".

The pair sheltered from the rain under a single black umbrella after they stepped out of the presidential black Cadillac.

William, wearing a blue jacket, blue trousers and white shirt without a tie, apologised to the Obamas, saying: "Sorry about the weather".

Kate, wearing a teal and maroon dress, shook hands with the president and his wife, who was wearing a long camel coat over a lighter camel dress, before Mr Obama offered her the umbrella.

But the Duchess refused it, leaving the US leader to hold it over both his wife and Kate.

The five chatted briefly, laughing together several times, before moving to pose for photos without the umbrella.

As they chatted in the courtyard, William was heard to say: "You've had a long day?"

To which Mr Obama, wearing a black suit and open-necked white shirt, replied: "We've enjoyed it."

As they went inside for dinner, they found an eager Prince George awaiting them.

A Kensington Palace spokesman said: "Prince George stayed up to meet the president and first lady when they arrived at Kensington Palace.

"He was able to show the president and first lady a rocking horse - given to George when he was born - and a stuffed toy - given to George when Princess Charlotte was born - that had been previously given to him by president and Mrs Obama."

Earlier, the US leader spoke after having lunch with the Queen and the Duke of Edinburgh at Windsor Castle and presenting her with a 90th birthday present - a reminder of her many visits to America.

Speaking at the Foreign Office, Mr Obama said the Queen "is truly one of my favourite people".

And he added: "She's an astonishing person and a real jewel to the world, not just to the United Kingdom."

Mr Obama said he had returned to the UK to "wish her Majesty, the Queen, a very happy 90th birthday".

He added: "Earlier today Michelle and I had the honour to join her Majesty and his Royal Highness, the Duke of Edinburgh, as their guests at Windsor Castle, where we conveyed the good wishes of the American people."

The president spoke fondly of the meeting, joking that he had "never been driven by a Duke of Edinburgh before", adding that the drive in the grounds of Windsor Castle had made for "very smooth riding".

He went on: "As for her Majesty, the Queen has been a source of inspiration like so many people around the world.

"She is truly one of my favourite people, and should we be as fortunate to reach 90, may we be as vibrant as she is."

The president and the first lady presented the Queen with a photo album chronicling her official engagements with US leaders and their wives.

Over the past decades Britain's head of state has made numerous trips across the Atlantic, from her first in 1951, as Princess Elizabeth, which saw her tour George Washington's historic Mount Vernon estate and meet President Harry S Truman.

She also visited Ground Zero in New York in 2010, the site of the 9/11 terrorist attacks, where she paid her respects.

The Obamas have a warm relationship with the Queen and her family and were invited to have lunch with the monarch and the Duke of Edinburgh at Windsor Castle, ahead of talks the president held with the Prime Minister.

The Queen is well used to hosting American leaders - Mr Obama is the 12th US president of her reign and the Queen has met all of them, except one - Lyndon B Johnson.

Philip showed the US leader and his wife the personal touch when he drove them and the Queen to Windsor Castle after the Marine One presidential helicopter had landed close to the monarch's Berkshire home.

The gesture was akin to picking up guests from the airport although the journey only lasted a few minutes, but Mr Obama looked delighted when he discovered Philip would be driving them.

The Queen, who wore a powder blue jacket and matching skirt, and Mrs Obama, reportedly dressed in a Oscar de la Renta print dress and Narciso Rodriguez coat, sat on the rear seats.

The Duke may be approaching his 95th birthday in June but he looked composed at the wheel of the Range Rover as it made its way around the castle's quadrangle and stopped outside the sovereign's entrance.

Inside the foursome posed for a photograph in the castle's Oak Room, in the Queen's private apartment.

She uses the room to hold audiences and in past years has recorded her Christmas broadcast there.

The private lunch lasted around an hour before the Queen and Philip said goodbye to the outgoing US leader and his wife.

In an ITV documentary to mark the Queen's 90th birthday, Mr Obama praised the Queen for her "generous human touch'' as he credited her with helping to ensure the lasting special relationship between the UK and America.

In 2009, after a photocall for G20 leaders at Buckingham Palace, the Queen and Michelle Obama were seen putting their arms around each other's backs.

A Buckingham Palace spokeswoman described it at the time as a "mutual and spontaneous display of affection and appreciation" between the two women.

They formed a strong friendship, bonding over a shared interest in the countryside, gardening and clothes.

As they said goodbye at the end of the visit in 2009, the Queen was overheard asking Mrs Obama: "Now we have met, would you please keep in touch?"

A successful state visit to the UK followed in 2011, and Mrs Obama has also returned to London privately with her daughters, Sasha and Malia, when they were given a guided tour of Buckingham Palace by palace staff.

Prince Harry has a close friendship with Mrs Obama in particular - through his support of wounded military personnel and his work with the Invictus Games, a Paralympic-style championship for injured military personnel from across the globe. Mrs Obama is a keen supporter of forces' families in the US.