Flanagan was born in Stockport, Cheshire. After leaving Grange Comprehensive School, in Oldham, Greater Manchester, he worked as a joiner. At 16 he went to London, taking on occasional odd jobs and for a time slept on the streets. Anthony has described this as a "very bad period." Anthony decided to become an actor after seeing a friend of his sister perform at RADA.
After studying acting at Oldham College, Anthony's first professional job was in a lead at the RSC in the 2001/2 season, playing Pat hartford in Peter Whelan's play A Russian in the Woods. This led to a collaboration with writer and theater critic Richard Edmonds on a one-man show on the life of Baudelaire entitled King of a Rainy Country.
Notable work includes roles in Vincent, Cracker, Life on Mars, Foyle's War, All About George, Casualty, Holby City, The Bill, Spooks, Spooks: Code 9, Doctor Who, Survivors, and Red Riding. He played his first major lead role in the short serial Instinct, broadcast in February 2007 and was due to appear in the West End in Glengarry Glen Ross in October 2007, but withdrew due to personal reasons.Anthony appeared in the BBC TV docudrama Heroes and Villains as Spartacus, first boratcast in 2008. He breaks the character of virile Roman hero and appeals full of doubt, irascible and Shakespearean. The series was successful in Europe and the USA, and was noted for the credibility of its historic sources. He took on the role of villain Dexter in ITV's Surivors. In 2009 he played the dogged but doomed journalist Barry Gannon in Red Riding. Initially a Channel 4 series, the trilogy has been successful all over the world, and regularly on the European film festival curcuit, winning many awards. Anthony was scheduled to appear in the film Acts of Godrey, which filmed in early 2010; however, he didn't appear due to theater commitments. He briefly appears in Glorious 39, a UK production written and directed by Stephen Poliakoff.
2010 was the year of Anthony's theater comeback. The ex-RSC came to the Birmingham Rep Theatre for a new adaption of Chekov's Cherry Orchard beside Josie Lawrence, where his performance was very well received.
In 2011, Anthony played the role of Steve in Life on the Line, a short film by Jamie Donaghue.
He also played in ITV's BAFTA award winning drama Appropriate Adult as Mike playing opposite Emily Watson who won a BAFTA for best Leading Actress for her role as Janet Leech. In 2012 Anthony appeared as Fergus in Being Human season 4. He then appeared as Andy in Julie Gearey's Prisoners' Wives, a BBC1 drama about how women deal with life after their husbands and families are sent to prison.
Also in 2012 Anthony returned as a Detective Chief Inspector in BBC1's Silent Witness to play D.C.I. Ronson.
In 2012 Anthony worked and appeared in three films. He plays the recruiting sergeant in Private Peaceful. Adapted for the screen by Simon Reade based on Michael Morpurgo's novel of the same name and was directed by award winning director Pat O'Connor. He appears in Kadri Kousaar's film The Arbiter as Mickey. He played the title role in Klaus Huttmann's feature film The List.
Anthony plays Arnold Hankin, a self-made successful businessman, in The Village. The six-part series was directed by Antonia Bird, who directed Anthony once before in Cracker, and also by Gillies MacKinnon. The first episode is set in 1914 and the complete series was written by BAFTA-winning writer Peter Moffat. The Village was broadcast on BBC1 in 2013. Anthony can be seen in the new series of Jimmy McGovern's Moving On and in the new season of Shetland.