He later surmised that the white horse in question represented the Saxon Steed in the coat of arms of the Elector of Hanover, the future King George I of Great Britain, his opposition to whom would later cause him much trouble.

He was closely associated with the radical Tory leader Lord Bolingbroke and was privy to the attempts made to bring about a Jacobite restoration on the death of Queen Anne (1702–1714).

At the start of the reign of the Hanoverian King George I (1714–1727), Bolingbroke fled into exile in France to join the court of the Old Pretender, and Wyndham took his place in England as the leader of the Jacobites.

Most ministers were inclined to agree to this for fear of offending a person of such consequence, yet Lord Townshend, Secretary of State for the Northern Department, in the belief that the government needed to show firmness, moved the motion to have him arrested.

Ten minutes of silence ensued while the other ministers considered their own responses, and finally two or three others seconded the motion and the arrest was decreed by the king, who on retiring to his closet took Townshend's hand and told him: "you have done me a great service today".

[5] Lord Stanhope brought down to the Commons a message from the King, desiring their consent for apprehending six members of their House on a charge of "being engaged in a design to support the intended invasion of the kingdom",[6] namely[7] Sir William Wyndham, Sir John Pakington, 4th Baronet, Edward Harvey (MP for Clitheroe),[8] Thomas Forster, John Anstis, and Corbet Kynaston.

[9] He was awoken at 5 in the morning and on searching his bedroom the colonel found incriminating papers in his waistcoat pocket, which listed his co-conspirators who planned to invade England and place the Old Pretender on the throne.

[2] This caused the king to circulate a hand-bill headed "Proclamation for apprehending Sir William Wyndham, Baronett", dated 23 September 1715, which offered a huge reward of £1,000 for his capture.

This was perhaps due to the fact that his father-in-law the 6th Duke of Somerset became a founding governor after his second wife, Charlotte Finch (1711–1773), became the first to sign the petition to King George II of its founder Captain Thomas Coram.

On her brother's death in 1750 she became (with the 7th Duke's only daughter Lady Elizabeth Seymour and her husband Sir Hugh Smithson, 4th Baronet) one of two co-heirs to the vast estates formerly belonging to the ancient Percy family, former Earls of Northumberland, including of Egremont Castle in Cumberland and of the jewel in the crown Petworth House in Sussex, rebuilt in palatial style by her father the 6th Duke, whose first wife had been the great heiress Lady Elizabeth Percy (1667–1722), only daughter and sole heiress of Joceline Percy, 11th Earl of Northumberland (1644–1670) of Petworth House and Alnwick Castle in Northumberland.