First of all, recognize that most lawsuits do not look like this one. The typical lawsuit involves an entry of a money judgment by a plaintiff against a defendant, or an an order granting a plaintiff possession of property in the possession of a defendant, or a defense judgment denying a plaintiff any relief. Usually this is for a breach of contract, or a civil wrong (called a tort), or based upon the property rights of the parties.
In those cases, the general rule is that a trial court judgment in favor of a plaintiff takes effect allowing the plaintiff to collect money or seize property from the defendant unless the judgment is stayed pending appeal. But, if the defendant does not stay (i.e. put on hold pending appeal) the trial court judgment (which often requires posting a hefty bond with the court), and the defendant does prevail on appeal, the defendant is entitled to restitution of the money or property taken by the plaintiff from the defendant pursuant to the reversed judgment. A defense verdict leaves nothing to stay, the pre-lawsuit status quo remains in place.
The rule has always been more complicated in equitable and public law cases such as the charitable trust case you mention, or, for example, contested election or a case regarding child custody.
On one hand, there is not the same right to stay a judgment pending appeal by posting a supersedes bond that there is in cases involving money or property. On the other hand, the appropriate court is more likely to stay enforcement of a judgment pending appeal in such cases without the posting of a bond, or, for example, by placing a caretaker trustee in charge of the purportedly charitable trust pending appeal. If a trial court denies a stay on appeal, an appellate court will ordinarily consider if that decision was appropriate on an expedited basis before considering the merits.
The trustees in a charitable trust case, if they were the parties appealing, would attempt to act consistently with the trust's claimed charitable status so much as possible consistent with the court's order, and the mere fact that a trust was not charitable would ordinarily not mean that the grantor of the assets in the trust would be entitled to their return. But, this could depend upon the reason that the trust was not found to be charitable.
A charitable trust might be found to be invalid because its assets were received in a fraudulent transfer from an insolvent grantor, in which case the creditors of the grantor would have the right to seize its assets, subject to restitution if the judgment were reversed on appeal.
A charitable trust might be found to have been operated in an uncharitable way, in which case the remedy might be to transfer the trusteeship of the remaining assets to a neutral third-party experienced charitable trustee, who might also be empowered to claw back donations made for uncharitable persons from the recipients or the former trustees.
A charitable trust might be found to have purposes that do not qualify as charitable, in which case it might become a non-charitable private trust that is still valid but doesn't receive the tax treatment it would as a charitable trust.
Also, sometimes winning on appeal is a moot point.
For example, in the U.S. case where public accounting firm Arthur Anderson was convicted of certain crimes in a trial court, this crime forced the entire enterprise into bankruptcy since it was disqualified from most of its contracts and lost the faith of those clients who didn't have to terminate its services, that its subsequent acquittal of all or most charges on appeal was irrelevant. The entity was dead and a financial disaster, and the appellate ruling meant almost nothing except for a few insurance companies insuring its officers and directors.
Similarly, appeals of election law issue decided after the disputed part of an election are conducted based upon a trial court ruling and any stay of that ruling imposed by the relevant courts, is irrelevant for the current election, even though they may clarify the rules in future elections.
Similarly, an appeal of a child custody decision about where the child goes to middle school next year might be moot by the time it is resolved on appeal.
In short, there isn't a clean general answer in a complex case like this one, even though there is a clean general answer that applies to the lion's share of civil cases that are pending on appeal. In a complex case like this one the outcome is highly fact specific.