Would it be a problem if the Nuvaring was stored at room temperature?

No - that is fine. The dispenser needs to store the NuvaRing in the fridge but you don't. They can stay stored in a cupboard or drawer, out of direct sunlight, for 4 months.

This from https://www.drugs.com/pro/nuvaring.html

"Storage:
Prior to dispensing to the user, store refrigerated 2°–8°C (36°–46°F). After dispensing to the user, NuvaRing® can be stored for up to 4 months at 25°C (77°F); excursions permitted to 15°–30°C (59°–86°F) [see USP Controlled Room Temperature]. Avoid storing NuvaRing® in direct sunlight or at temperatures above 30°C (86°F). For the Dispenser: When NuvaRing® is dispensed to the user, place an expiration date on the label. The date should not exceed either 4 months from the date of dispensing or the expiration date, whichever comes first."