As I understand it, the monarchy doesn't own otherwise public lands, in fact it's exactly the opposite - they lease their private lands to the government to be used as public lands[0]. In other words: this is entirely private land that has been granted to the public.
"Historically, Crown Estate properties were administered by the reigning monarch to help fund the business of governing the country. However, in 1760, George III surrendered control over the Estate's revenues to the Treasury,[5] thus relieving him of the responsibility of paying for the costs of the civil service, defence costs, the national debt, and his own personal debts. In return, he received an annual grant known as the Civil List."
"Historically, Crown Estate properties were administered by the reigning monarch to help fund the business of governing the country. However, in 1760, George III surrendered control over the Estate's revenues to the Treasury,[5] thus relieving him of the responsibility of paying for the costs of the civil service, defence costs, the national debt, and his own personal debts. In return, he received an annual grant known as the Civil List."
[0]: https://en.wikipedia.org/wiki/Crown_Estate