The Duchess of Cambridge has said she “can’t wait to meet” the Duke and Duchess of Sussex’s daughter, whose naming has sparked controversy.

Kate was quizzed about Harry and Meghan’s baby Lilibet at the end of a visit to a Cornish school with US first lady Jill Biden.

The two women met for the first time at Connor Downs Academy in Hayle in the county of Cornwall, where world leaders have gathered for their G7 summit.

A broadcast journalist from an American network put Kate on the spot when she asked: “Your Royal Highness, do you have any wishes for your niece Lilibet?”

The duchess replied smiling: “I wish her all the very best. I can’t wait to meet her because we haven’t yet met her yet, so hopefully that will be soon.”

She was also asked if she had spoken with Meghan: “Have you FaceTimed with her?”

Kate replied “No, I haven’t.”

White House aides and Kensington Palace officials moved swiftly to end the impromptu press conference, asking journalists to leave the school room where the duchess and Mrs Biden had been taking part in a round-table discussion with experts on the importance of the early years of childhood for future outcomes.

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Kate and the US First Lady during their visit to a Cornish school (Aaron Chown/PA)

Harry and Meghan’s lawyers have written to the BBC after the corporation claimed in a report the Queen had not been asked about naming their baby Lilibet, which was the monarch’s childhood family nickname.

It has been widely reported that Harry and Meghan did consult the Queen before announcing the name of their baby, who was born in California on Friday June 4.

A spokeswoman for the Sussexes said their daughter’s name was mentioned by Harry in a conversation with the Queen and “had she not been supportive, they would not have used the name”.

Kate and the US first lady share a passion for improving early years development and displayed their own informal transatlantic alliance on education during the visit.

The duchess told Mrs Biden: “I’m committed to this for a long time. I hope our two countries can continue on this and share data.”

Kate, wearing an Alexander McQueen outfit, and Mrs Biden, an English professor and former high school teacher, first visited the school’s reception class of four and five-year-olds led by the school’s early years specialist David Masters.

The two women heard how the academy’s pupils are supported through a bespoke early years foundation stage designed to help them develop as independent, confident and successful learners.

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Mr Biden and the duchess joined children in a classroom (Aaron Chown/PA)

A US journalist asked how important the first five years of a child’s life were. “Very important. It’s the foundation of everything,” Mrs Biden replied.

“This is amazing to see what these children are doing and how far advanced they are at four and five years old.”

Pupils at the school are also taught to value animal welfare and go out every morning to look after a collection of rabbits and rescued battery hens in the grounds.

Kate and Mrs Biden were taken to see the rabbits – Storm, Wampa and Cindy – and the first lady went armed with a bowl of large carrots and stood alongside Kate, chatting over a gate to Mr Masters and four pupils with Storm.

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Kate and Mrs Biden watch during feeding time for the rabbits (Aaron Chown/PA)

The duchess and Mrs Biden then led a round-table discussion with a number of representatives from the early years sector who have helped influence Kate’s efforts to create a national conversation that gives every child in Britain a better start.

They were joined in the room by experts in developmental neuroscience, education and early years from the UK and US via a video call.

The duchess explained how her research into the long-term effects of addiction and family breakdown had convinced her to devote much of the rest of her royal career towards boosting the life chances of many children by improving their first five years.

She said: “Ultimately, my hope is we change the way we think about early childhood with the generations to come, and I am committed to this for the long term.”