Update(1730ET): It begins... and a White House statement is also expected imminently:

UK'S SUNAK AUTHORIZES JOINT MILITARY STRIKES AGAINST HOUTHIS

And the Times (UK) reported an hour ago:

Britain expected to join U.S in carrying airstrikes on Houthi military positions in Yemen on Thursday night — Times

The Houthis have said they are not scared of US and UK threats. While the US does not confirm future military operations before they happen, Reuters has the following details from the British side:

Britain is expected to join the United States in conducting air strikes on military positions belonging to the Iran-backed Houthis in Yemen "within hours", the political editor for the Times newspaper reported on Thursday.

British Prime Minister Rishi Sunak's Downing Street office did not respond to a request from Reuters for comment, while the Pentagon and the White House each declined to comment on the report.