Yes, it has always been owned by the Crown Estate.
The original Bagshot Lodge was built between 1631 and 1633, one of a series of small lodges designed for King Charles I by Inigo Jones. It was remodelled between 1766 and 1772 for George Keppel, the 3rd Earl of Albemarle. In 1798 it was again remodelled for the Duke of Clarence (later King William IV), who lived there until 1816. A later resident was Prince William Frederick, Duke of Gloucester, nephew of King George III; his widow, Princess Mary, daughter of King George III, continued to live there after his death until she moved out in 1847. The original house was demolished in 1877-78.
A new building with 120 rooms was built in 1879 by Queen Victoria for her son Prince Arthur, Duke of Connaught and Strathearn. The Duke died at Bagshot Park in 1942. The house thereafter served as regimental headquarters and depot of the Royal Army Chaplains' Department. They vacated the building shortly before the Earl and Countess of Wessex took over the tenancy from the Crown.