Conway married in 1747 Caroline, daughter of General Campbell (afterwards duke of Argyll), and widow of the earl of Aylesbury.
Other experiments in inductive telegraphy were made by Preece, aided by the officials of the British Postal Telegraph Service, in Glamorganshire in 1887; at Loch Ness in Scotland in 1892; on Conway Sands in 1893; and at Frodsham, on the Dee, in 1894.
Henry Seymour Conway's elder brother, Francis, 2nd Baron Conway, was created marquess of Hertford in 1793; his mother was a sister of Sir Robert Walpole's wife, and he was therefore first cousin to Horace Walpole, with whom he was on terms of intimate friendship throughout his life.
Having entered the army at an early age, Conway was elected to the Irish parliament in 1741 as member for Antrim, which he continued to represent for twenty years; in the same year he became a member of the English House of Commons, sitting for Higham Ferrers in Northamptonshire, and he remained in parliament, representing successively a number of different constituencies, almost without interruption for more than forty years.
In 1757 he was associated with Sir John Mordaunt in command of an abortive expedition against Rochfort, the complete failure of which brought Conway into discredit and involved him in a pamphlet controversy.