Greswell was born at Denton, Lancashire, on 22 July 1800, the fourth son of the Rev.
For many years he devoted the proceeds of his tutorship to public and charitable objects, his personal expenses being defrayed from a modest fortune brought by his wife, Joana Julia Armitriding, whom he married in 1836.
Many kindly and beneficent acts are related of Greswell, whose 'chief characteristics were great and varied learning, boundless benevolence, and a childlike simplicity'.
His daughter, Joana Julia Greswell, published at Oxford in 1873 a 'Grammatical Analysis of the Hebrew Psalter.'
Greswell's publications were a paper On Education and the Principles of Art, 1843, and a Memorial on the Proposed Oxford University Lecture-rooms, Library, Museums, &c., 1853.