Absolutely anybody can own greenbelt land - farmers, householders whose house is on greenbelt land, companies, the NT, the MOD and so on. Very little land is actually owned by government departments (apart from the MOD). You could buy a piece of land in the greenbelt - and you would be free to use it for what it is currently classified for, no problem.
Greenbelt is generalised term for planning policy classification. It ensures that land which is regarded as sacrosanct from change in its current usage has a policy applied to it locally that discourages a change in its use.
In rare situations planning applications are approved for development of 'greenbelt'.