Moore is perhaps best known for his role in the theater adaptation and subsequent film version of The History Boys. His other film credits include Goodbye Christopher Robin. He portrayed King Edward VIII in Wallis & Edward.
He starred in a 2010 TV adaptation of Ben-Hur alongside James Faulkner and Hugh Bonneville. He appeared in Julian Fellowes's 2012 miniseries Titanic, which also starred Jonathan Howard and Maria Doyle Kennedy. He also starred in the 2019 series War of the Worlds, alongside Elizabeth McGovern.