Wickham was the oldest son of Joseph Wickham and Abigail Parker of Cutchogue, Long Island, New York. Wickham inherited nearly all of his father's large estate at age 22, including the legendary Robins Island. He lived in the Old House in Cutchogue, which was built in 1649. Currently a museum, it is said to be one of the oldest English-style houses still in existence in the USA. Wickham married Mary Goldsmith and had several children. He was the brother (and rival) of Patriot leader Thomas Wickham and the uncle of noted Federalist attorney John Wickham.
Wickham was elected to the position of Fence Viewer and Prisor of Damage in 1751. In 1754 and 1755, he was elected Overseer of the Poor and he was appointed justice for the County of Suffolk in 1763. He was elected town assessor in 1765 and 1766, then elected nine times to the highest post of local government, town supervisor of the Town of Southold. He was also commissioned a major in the local militia.
During the American Revolution, Wickham was known for his pro-Loyalist views. On December 13, 1777, he was kidnapped by Connecticut rebels while serving out his term in elected office. He was paroled shortly thereafter, but on October 22, 1779, New York's legislature passed a bill of attainder, requiring him to forfeit his property without compensation and banishing him from the state under threat of death. Wickham was forced to move to Connecticut, where he died shortly thereafter. He insisted he was innocent of the charges, but was never granted a trial. Acts of attainder were banned under the U.S. Constitution, which was adopted a few years after Wickham's death.
Shortly before his death in exile, Wickham wrote:
In 1989, several of Wickham's heirs filed a lawsuit to try to regain ownership of Robins Island because its owner at the time tried to develop the island. The lawsuit failed, but also discouraged development. Most of the island is now protected by an easement to The Nature Conservancy.
Wickham was interred in the Raymond Cemetery in Waterford, Connecticut.