Opened by Samuel Greg & Co. in 1822, the mill was taken over by Storey Bros. (one of Lancaster's leading firms) in 1862. Storeys ceased trading in 1982, and after a period of dereliction then Council-funded restoration the mill became the UK headquarters of Reebok, the sports/footwear company, in 1990.
I presume the tower contained water, maintaining supply at a constant pressure.