Gower was born in Aintree, Merseyside, near Liverpool in North West England. His early schooling included Davenhill Primary School in Aintree and Great Sankey High School in Cheshire. It was during Year Nine of his studies that Gower began acting, appearing in school plays and musical productions. He enrolled in Barrow Hall College, Great Sankey High School's sixth form college, in 2007, and while studying for his A levels joined a local amateur drama group. Gower, an Oxford School of Drama graduate, was awarded the prestigious Spotlight Prize for Best Actor in July 2010.
March 2011 saw Gower's British television debut, on the ITV series Monroe, in the role of Dr. Andrew Mullery, a general surgeon in training. To research the role, Gower watched two coronary artery bypass surgeries while standing next to actual surgeons. Gower appeared in both series, reprising his role as the newly promoted General Registrar, under Head of Clinical Services Alistair Gillespie (Neil Pearson), in the second series.
That same year, Gower appeared in the television broadcast of the musical Frankenstein's Wedding... Live in Leeds, a modern adaptation of Mary Shelley's gothic novel Frankenstein. Instead of being pre-recorded as a traditional television musical, it was broadcast live, with an audience of 12,000 in attendance at Kirkstall Abbey, on BBC Three.
In 2012, Gower joined the cast of BBC Three's Being Human, as vampire solicitor Nick Cutler, in series 4 of the hit television series. Filming for the series began in Barry, Wales in late July 2011 and was broadcast in 2012. Gower was the recipient of critical praise for his work on the series. This year also saw a guest role in series 4 of E4's science-fiction comedy Misfits and a run as Seiffert in Manfred Karge's surreal play The Conquest of the South Pole at the Arcola Theatre. In his first radio performance, Gower played Lupin in the BBC Radio 4's comedy Diary of a Nobody, adapted by Andrew Lynch from George and Weedon Grossmith's Victorian novel.
2013 saw a return to episodic television for Gower. He appeared as Sherlock Holmes in series 6 of CBC's Victorian drama Murdoch Mysteries and would later reprise his role in series 7. After a cameo in Showtime's The Borgias, Gower appeared as Lord Strange in several episodes of the BBC's mini-series The White Queen.
In 2014 Gower was featured in Jon Stewart's film directorial debut, Rosewater, as Jimmy the Avid Editor. That same year, he appeared in series 2 of the BBC's drama The Village as Gilbert Hankin and in series 2 of ITV's crime drama Endeavour as Nicholas Myers.
The next year (2015) Gower was cast in the role of Caligula in NBC's drama A.D. The Bible Continues. His next project was the character Mark, an ambitious young banker, in the three-part BBC series Capital, based on John Lanchester's novel of the same name.
In 2016, Gower appeared as Winston Smith in a theatrical adaptation of George Orwell's 1984 at the Playhouse Theatre in London. He also appeared as Prince Charles Stuart in Season 2 of the Starz historical dramatic series Outlander, based upon Diana Gabaldon's best-selling book series.
Gower is currently working on his first documentary, an in-depth look at Merseybeat musician Tommy Quickly, with friend and fellow actor Stephen Walters. The project was funded through Kickstarter, a widely used crowdfunding platform.
Gower is a member of the rock band Emerson. A singer and songwriter, he is a self-described "obsessed" Beatles fan. The band has played The Cavern Club (the Liverpool club where The Beatles were discovered) several times.