Leslie Ramos, the author of a forthcoming book on arts giving, said that Britain “just doesn’t have the culture of philanthropy like the U.S., especially for the arts.” Several major patrons died recently, she added, and younger donors were not filling the gap. They prefer to donate to social justice causes, or organizations fighting climate change, she said.

Paul Ramsbottom, the chief executive of the Wolfson Foundation, one of Britain’s largest institutional arts donors, which gave around $630,000 toward the National Portrait Gallery renovation, said that funds like his were seeing a “rising tide” of applications that they couldn’t possibly satisfy.

This increasing reliance on donors comes as several major British museums embark on multiyear overhauls. The British Museum is expected to soon announce a renovation that The Financial Times has reported will cost £1 billion, around $1.3 billion. The National Gallery has also been trying to raise £95 million for a refurbishment. In May, Anh Nguyen, the museum’s director of development, told an audience of donors and reporters that trying to secure the money had given her “sleepless nights” and “heart palpitations.”

Cullinan, the National Portrait Gallery director, said the key to grabbing donors’ attention was having a compelling project. Before the renovation, the National Portrait Gallery — founded in 1856 with the idea of displaying portraits of the most eminent people in Britain — was a much-loved institution, he said, but it had obvious room for improvement. Visitors could easily miss its former entrance, a small doorway on a busy street. Inside, he added, the museum’s corridors often felt like a rabbit warren and some of its displays “hadn’t been touched for 30 years.” Its only educational space was “in a dingy basement,” he added.

Its displays were not representative of contemporary Britain, Cullinan said: Just 3 percent of portraits on the walls were of people of color. (After the refurbishment, that has increased to 11 percent.)