Brammer performed as 'Chris' in the Miss Saigon 25th anniversary performance.
Originally from Exmouth, Devon, Brammer is the youngest of seven children.
In 2008, while a student at Exeter College, he competed in Any Dream Will Do, and made it to the final 20 contestants, although he was not selected to be on the show.
However, Any Dream Will Do judge Bill Kenwright asked him to appear in the 2008–2009 UK tour of Joseph and the Amazing Technicolor Dreamcoat, where he played Zebulon and understudied both Benjamin and Joseph.In 2009, he made his West End debut as Marius in Les Miserables, where he spent 18 months in the role.
He said of Les Mis, "I should be sick to death of the show, but I can't get over it.
Since that time, he also appeared as Billy in the 2012 revival of Taboo at the Brixton Clubhouse, and until 15 March 2014 was playing Billy and understudying Albert in the West End production of War Horse.On 21 November 2013, it was announced that he would be starring as Chris in the West End revival of Miss Saigon.
Previews began on 3 May 2014.
He revived his role as Chris in the Broadway revival of Miss Saigon which began previews on 1 March 2017.
He remained with the Broadway revival until its closing on 14 January 2018.
On 5 December 2015, Brammer made his first appearance as Jack Diamond, the receptionist, in Casualty.
In December 2016, Brammer played the role of Oliver in the Finale special of Vicious.
Oliver served as main characters Freddie and Stuart's brand new upstairs neighbour following the departure of Oliver's flat's previous occupant Ash (Iwan Rheon), whom Freddie and Stuart were close to.
From 22 July 2019, Brammer will play the role of Fiyero in the West End production of Wicked.