No, because there is no public land and no politician is ever going to get very far trying to advocate using public funds to purchase private property for the purpose of creating public hunting access.
National parks are private farmland and forestry where public access is allowed by law, but even then there are restrictions. The hunting rights are private.
Forestry Commission is I believe a public body set up for commercial timber production but with private ownership rights. Some land is owned, but some is only leased from landowners. Where FC owns sporting rights, they use their own Rangers for crop (tree) protection and they also let sporting rights.