Norcott was born in Westminster on 3 August 1777. His father, Lieutenant Amos Norcott of the Green Horse Regiment - the 5th Dragoon Guards - had already shipped out for Barbados and died en route. The son grew up knowing his father only as a story. He entered the British Army in 1793 at sixteen, taking a second lieutenancy in the 33rd Foot, the regiment he would forever after share with the future Duke of Wellington. He served on the staff of his great-uncle, Robert Cuninghame, Baron Rossmore, who was then Commander-in-Chief of Ireland. From his earliest army days, Ireland was part of his life - first as a place where he reported to a great-uncle, ultimately as the place where he would die.

The 33rd Foot sent Norcott to India, and India is where he became friends with Arthur Wellesley. The friendship would last. When Norcott ran up gambling losses, Wellesley settled them. In 1802 Norcott transferred to a regiment that did not yet quite exist: the newly formed 95th Rifles, the green-jacketed marksmen who would invent modern light infantry tactics across the next decade. With the 95th he fought through the Peninsular War, where Wellesley - by then Lord Wellington - commanded the army. Norcott was wounded at the Battle of Corunna in January 1809, the campaign that ended with the British evacuation under fire and the death of Sir John Moore. He recovered. At Waterloo on 18 June 1815, Major Norcott commanded a battalion of the 95th in the line. He was wounded again during the battle. He survived this too.

Promotion came slowly in the long peace after Waterloo. Norcott's lieutenant-colonelcy was confirmed in September 1819 with the brevet rank of full colonel of the 8th Foot Regiment. He reached major general in July 1830. On 13 September 1831 he was knighted at St James's Palace as a Knight Commander of the Royal Hanoverian Guelphic Order. In 1834 the Crown sent him to act as Governor of Jamaica - a colonial appointment that placed him in the middle of one of the most charged years in Caribbean history, just after Britain had passed the Slavery Abolition Act and the apprenticeship system was being imposed on Jamaica's formerly enslaved population. The work belonged to a colonial system whose human cost would echo for generations after Norcott left the island.

Norcott returned to Britain and was given command of the Cork District, a major garrison area in the south of Ireland that included the gunpowder mills at Ballincollig and the network of forts ringing Cork Harbour. He took up residence at Marysboro House near Ballincollig and died there on 8 January 1838 at the age of 60. His family carried the soldier's life forward in three directions. His son Robert went into the army and died of cholera in India. His son William Sherbrooke Ramsey Norcott became a lieutenant general of the Rifle Brigade, fought in the Crimean War, and served as aide-de-camp to Queen Victoria. His son Charles Rossmore Robert Norcott became a superintendent of the Western Australia Police and acted as aide-de-camp to his father in Cork - and died, strangely, only six weeks after his father, in early 1838. Three generations of Norcotts, scattered from Cork to Crimea to Western Australia, all setting off, like the first Amos Norcott had, on ships.