Source: Michael Doyle Their Name Liveth For Evermore: The Great War Roll of Honour for Leicestershire and Rutland.
He was the son of Arthur Edward Routledge, an engineering pattern maker, born 1860 in Carlisle, Cumberland and his wife Grace Ann Routledge (nee Perkins, married in the 4th quarter of 1884 in the Lambeth, Surrey district), born 1865 in Plymouth, Devon. William Arthur, a schoolboy, was born in the 4th quarter of 1886 in Lambeth, Surrey, his siblings were, Mabel Grace, a schoolgirl, born 1886 in Lambeth, Maud Elizabeth, a schoolgirl, born 1887 and Herbert G., born 1890, the latter two siblings were both born in Battersea, Surrey, in April 1891 the family home was at Ingrave Street, Battersea. In March 1901 William was employed as an engineering electrician and fitter and was residing in the family home at 43, Ratcliffe Road, Loughborough, Leicestershire together with his father an electrical engineering pattern maker, his mother and siblings, Mabel, a hosiery trade worker, Maud, a hosiery trade warehouse worker, Herbert, Dorothy, born 1892 in Battersea, Reginald, born 1894, Alice May, born 1897 and John Robert, born 1900, the latter three siblings were all born in Loughborough. In April 1911 William was absent from the family home at 12, Ratcliffe Road, Loughborough, residing there was his father, an electrical engineering pattern maker, his mother and siblings, Mabel, a hosiery trade blouse machinist, Reginald, a grocery trade assistant, Alice, a hosiery trade warehouse worker and John, a schoolboy, also residing in the family home were William’s nephew, Walter Routledge, born 1908 and niece, Ethel Routledge, born 1910, both children were born in Loughborough.
The War Diary records: 22 Oct-14 - The Battalion had relieved the West Yorkshire Regiment the previous day at the Chemical Factory at Rue du Bois, they held the trenches and were heavily shelled by shrapnel and heavy Howitzers all day, casualties were Lieutenant Prain and Dods killed, Lieutenant Smeathman was wounded and died on the 24th October, eleven Other Ranks were killed and twenty five wounded. William was one of seven Leicestershire men who lost their lives on this day.
On Saturday January 9th 1915 The Leicester Chronicle and Leicestershire Mercury published the following article under the heading. “LOSSES IN THE LEICESTERSHIRE REGIMENT.” – The following casualties in the Leicestershire Regiment are announced. – KILLED. Corporal W. A. Routledge.

Source: Leicestershire War Memorials Project.
The following information was kindly provided by John Grainger, from his book 'Blue Line to Trench Line' in 2017.

William Arthur Routledge was born 21st September 1886 at 60 York Street, Lambeth. Birth was number 153 of 1886 in Sub-District of Waterloo, in Registration District of Lambeth in the County of Surrey. Birth registered by his father on 30th October. On certificate parents are Arthur Edward & Grace Ann (formerly Perkins). Father's occupation pattern maker. Routledge attested for service with the Leicestershire Regiment on the 20th June 1904 as Private 7372 and was posted to the 1st Battalion. In 1911 he was serving as a Lance Corporal at Fort George, Madras, India. He was recorded as 25 years of age and his place of birth, Battersea, London.
On the 25th May 1912 he was appointed to Durham County Constabulary as PC 830, 3rd class constable. In June 1912 he was posted to Bishop Auckland. 1.12.1912 Appointed 2nd class constable. 1.6.1914 Appointed 1st class constable. June 1914 Transferred to Castle Eden Division. 4.8.1914 Recalled to the colours and KIA 22.10.1914. His name is commemorated on the war memorials of Durham County Constabulary and Castle Eden.