Elcoat was born in the small village of Elton but lived most of his life in Stockton-on-Tees.
In 1895 he was appointed secretary-manager of Stockton after the incumbent was sacked for failing to provide documents to the club's accountant.
Between 1986 and 2011 it had been mistakenly accepted that William's younger brother George was Woolwich Arsenal's manager.
Elcoat's tenure at Woolwich Arsenal was quite short - he resigned on 20 February 1899 citing difficulties with the club's board of directors.
This biographical article related to an English association football manager is a stub.