The Queen of England wouldn’t stand for this. Australia had a government shutdown in 1975 and the Governor General threw them all out and called an election.
Australia’s 1975 shutdown ended pretty differently, though, than they do here in America. Queen Elizabeth II’s official representative in Australia, Governor General Sir John Kerr, simply dismissed the prime minister. He appointed a replacement, who immediately passed the spending bill to fund the government. Three hours later, Kerr dismissed the rest of Parliament. Then Australia held elections to restart from scratch. And they haven’t had another shutdown since.
The full story for some fun reading.