William was educated at Oakham and Newark grammar schools, and in 1714 he was articled to Mr Kirke, attorney at East Markham, in Nottinghamshire.
In 1834 he was articled to a solicitor in Bury St Edmunds, but the uncongenial and sedentary employment soon broke down his health.
Having passed his law preliminary, he was articled to a firm in Portmadoc, and in 1884 obtained his final qualifications.
His father, Francisco Antonio Zumalacarregui, was a lawyer who possessed some property, and the son was articled to a solicitor.
In 1803 Palgrave was articled to a firm of solicitors, but was called to the bar at the Middle Temple in 1827.
At the age of nineteen, he was articled for five years as clerk to the master of a school in Spital Square, London, with whom at the end of that time he entered into partnership. In 1750 he read a paper before the Royal Society on a method of making artificial magnets, which procured him election as a fellow of the society and the award of the Copley medal.
He was articled as a law clerk in Edinburgh, and his Elegy on Craigmillar Castle (1776) was printed during his clerkship. In 1781 he removed to London to devote himself to literary work, publishing in the same year a volume of Rimes of no great merit, and Scottish Tragic Ballads.