[1] He is known for playing the title character in the BBC fantasy series Merlin (2008–2012), Leo Elster in Humans (2015–2018), and Billy Clanton in Kenneth Branagh's Belfast (2021).

[6] Morgan starred in several college productions including Surgical Sensations at St. Senapods and Bad Day at Black Frog Creek.

[10] That same year, Morgan played the role of Esteban, an aspiring teenage writer, in the Old Vic stage adaptation of Pedro Almodóvar's All About My Mother.

[11] The Independent noted Morgan brought Jimmy to life through "twisting, twitching mood-swings and mix of half-druggy cackling punk and half-angelic visionary."

"[12] In 2011, Morgan performed in the Royal Court Theatre production of Pedro Miguel Rozo's play Our Private Life to favourable reviews as Carlos, a "bipolar compulsive fantasist" homosexual who suspects he was molested by his father as a child.

[20] The Washington Post critic said that he "can't remember a better Ariel," and that Morgan's portrayal was "mysterious, in a way I've never quite experienced before – also wonderfully vulnerable".

[24] Morgan made his National Theatre debut in May 2018, starring in the new production of Brian Friel's Translations, directed by Ian Rickson.

[25] In 2019, he appeared in Arthur Miller's All My Sons at the the Old Vic as Chris Keller, a production that was screened in cinemas and later made available through the National Theatre's online streaming platform in 2021.

[29] In television, Morgan first appeared in 2007 as the embarrassed gay son John Leary in sketches on The Catherine Tate Show's "Christmas Special."

[34][35] After spending 2013 in back-to-back stage productions, Morgan reappeared on television as an ambitious journalist in an episode of the 1950s Irish crime drama Quirke, based on the books by John Banville.

[37] Morgan was approached to play the role, meeting with writer Allan Cubitt and lead actress Gillian Anderson in London to discuss his involvement.

[38] In June 2015, Morgan began his role as the mysterious fugitive Leo Elster in Humans, a British-American AMC / Channel 4 eight-part science fiction drama.

[41] In mid-2016, Morgan starred in the BBC One Victorian era ghost hunter series The Living and the Dead as the central character Nathan Appleby.

[53] To prepare for the role of the soldier, Morgan read Brittain's works, as well as personal letters exchanged between Vera, Victor, and their friends.

[55] In the fourth quarter of 2015, Morgan was featured as Frankie Shea, brother-in-law and personal driver of lead characterReggie Kray, in the 1960s London gangster film Legend.

[74][75] In 2022, he completed filming on an adaptation of Eugene O'Neill's Long Day's Journey into Night, an independent production that has faced financing challenges.

[78] In December 2014, he starred as Newton Pulsifer in the first audio dramatisation of the popular book Good Omens, written by Neil Gaiman and Terry Pratchett, and directed and adapted by Dirk Maggs, also on BBC Radio 4.

[79] On 4 December 2016, Morgan read Autumn Journal on BBC Radio 3, which was about Louis MacNeice's poetic testament of life in 1938, written against the turbulent backdrop of the Munich Agreement, the fall of Barcelona and Britain's preparations for an inevitable war.

[89]From October 2009 until its conclusion in March 2011, Morgan was an ambassador for the A Night Less Ordinary scheme in which Arts Council England offered free theatre tickets to anyone under 26 years old.

The process culminated in four world premieres presented to an audience of more than 1,000, and raised £182,000 towards the work that the theatre does with emerging talent, schools and the community.