In a sense, the answer to this is both everybody and nobody. The most valuable assets in the World Balance Sheet will be the global commons - the atmosphere, the ecosystems, the water and carbon cycles etc. Nobody owns them. Even where we have agreed property rights, eg on land, morally the property owners don't actually own the animals that visit the land or make their homes on it. They don't own the air above it or the sunlight and rain that falls on it.
What we need is a Global Governance Body ("GGB"), that would be the custodian of the World Balance Sheet on behalf of humanity. The main role of the GGB would be to ensure proper stewardship of the assets. Another role would be to ensure effective distribution of the essential supplies and services sustainably sourced and provided within the global welfare system, for example under a Global Welfare Insurance Scheme. A third role would be to regulate allowable trade within limits to ensure market forces don't compromise long-tern sustainability of the system of systems on Earth. So perhaps in another sense the GGB would be the owner of the World Balance Sheet, on behalf of us all.
Might the United Nations perform this function of GGB? Perhaps.
The Planetary CFO - working towards a sustainable World Balance Sheet.