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 UK Parliament clearly believes it is making the world a safer place by deciding to authorise air strikes in Syria by UK aircraft.  Therefore, under a global system relating to the World Balance Sheet, the costs of those air strikes, in terms of the equipment, staff and logistics etc, should be borne by those who are being made safe.  This should also include any damage to infrastructure, and some measure of non-combatants’ deaths, eg in terms of the loss of productive and valued members of the Syrian public.

If global welfare insurance existed, then it would be easier to charge the costs of security to those whose security and safety is being maintained, because their global welfare fund would cover the costs. To the extent that this is citizens of Syria, then a bill should be sent to the Syrian Government, who have failed to provide that safety so far.  To the extent that it is people in other countries, including Western nations, then they should also receive a bill. 

The United Nations could act as debt collector.  And if Governments like the Syrian one (which would have by far the largest bill) can’t pay, then assets of value in the World Balance sheet (eg sustainably managed land) could be seized by the UN in lieu of payment).

In this attached paper, I attempt to reconcile the views of George Monbiot and Dieter Helm on valuing nature, which is often a controversial topic.