I forgot this point.
What is registered in the UK might be the S.A. entity registered as a foreign "corporation" to do business in the UK.
This sort of scenario is quite common.
Thus you in fact have the same entity registered to do business in two jurisdictions and so doing business. The trick and benefit of this to the COS in light of the avoidance of the Wollersheim settlement would be that the UK assets taken over by the Ozzie domiciled entity could be "moved" or otherwise not attached if the existence of the Ozzie charitable entity was unknown to the Wollersheim attorneys and is not included as a party to the judgment in favor of Wollersheim.
Foreign business entities registered to do business in a different jurisdiction is quite common. Indeed, when I set up "Ability Consultants, Inc" it was originally a Delaware Corp., then registered as a foreign (alien) corp to do business in the State of New York.
The key to this is to read the registration dox at Companies House and see what they say . . . . it will reveal the domicile of "incorporation" . . . such dox actually state where the business entity is incorporated and the fact that it is a foreign corp registered to do business in the jurisdiction (whose registration one is viewing the dox of) when this is the case.