The work was sold at a Christies auction last year for $77 million on behalf of the Aberconway family of Wales, who inherited it in 1947 following the death of Samuel Courtauld.
The British government placed an export bar on the work in the hope a British buyer could be found and the painting could be kept in the country. However, the ban expired in December, the British broadcaster said.
"'Child with a Dove' is an iconic Picasso painting, and has a long history in British collections. It is one of Picasso's key early works, and marks a transitional moment in his career, the move into his much celebrated Blue Period," Lord Inglewood, chairman of the Arts Council-managed Reviewing Committee, told the BBC.
"It is a great shame that institutions could not raise the funds necessary to keep this beautiful piece of art in this country. But this is exactly why export bars are so very important in protecting the nation's cultural heritage. In the case of the Picasso, clearly money was the problem, and while steps are being taken to increase philanthropy in the country, this suggests they may not be enough."