You may have been an employee.
If the volunteer's activities look, walk and quack like an employee's activities, and would be a type of work that the nonprofit would hire to an employee but for the volunteer stepping up, then there's a fairly compelling argument that the volunteer is acting in the capacity of an employee, and is well aware of that at the time of work.
You would need to prove a) it was a freelance relationship in which it would be normal for you to retain copyright from a client, and b) that there was understanding that this was not volunteer work and you never intended to donate it. (rather hard to prove now).
If you enjoyed any tax benefits of volunteering.
If you deducted mileage, cost of donated materials, etc. It means those deductions are void, and you may have misfiled your taxes!
Such deductions also are evidence for the question of intent to donate vs retain.
You have no other use for it.
There's also another problem. If this was work that you could cheerfully license to another firm instead -- say you build a Web gift shop for them - and you could unplug this and plug it into your new organization, that might be one thing. But this logo and design work was tailored just for them. You had to know that this would be used by them, and you have no possible financial gain for reclaiming it, since it is unfitful for any other client.
Therefore the very fiction of your owning it is patently absurd; you went into it knowing it would be theirs. What use would you have for it (other than hurting the nonprofit by denying their use, or even less impressively, extorting it back to them; private benefit which is taxable.) I think any court would hold that if you wanted that relationship, you should have contractually provided for that before work begun.
Volunteerism is irrevocable.
We once had a volunteer quit. 10 years before he had built a cabinet to fit our radio chargers. He cut the power cord and took the hutch off the wall. Nuh-uh. What he overlooked is he did it using our wood, our saw, our shop space, and our varnish, and "our time" (in the sense that we would have redirected his time if that wasn't the best use of it, and he would've complied). Volunteerism becomes the property of the organization the day you do it.
If you don't want to do it, don't do it. There's no takes-backsies.
As a matter of policy, though...
The nonprofit should continue use for as long as it takes to make plans and replace it. They have no legal obligation to do so, but there's a big strategic win to honoring donor wishes. Those donors may calm down later, and respecting their wishes will go far toward them warming to you later.
When the volunteer sees his work disappear, he may have another thought about whether this was a smart thing to do. It's one thing to leave, and quite another to witness all signs of you ever having been there eradicated.