I've done a little digging around on the internet today for information on fly fishing in the Carpathians. Now that I know Ukrainian pretty well, this is a lot easier. It's still difficult though because many Ukrainian fly fishing sites and forums are in Russian, or a frustrating mix of Ukrainian and Russian - especially forums.
I'm kicking myself for not figuring out what the Ukrainian is for "fly fishing" earlier - нахлист. While fishing is wildly popular in Ukraine, fly fishing is not. When I explain how I like to fish, I usually say something about fishing for trout in mountain streams. If the conversation goes on longer I try and describe a fly. Most people nod with familiarity, but aren't particularly interested in fly fishing because A) trout are simply not present in most of Ukraine, and B) the fishing most Ukrainians have access to is warm-water, for carp and related species. Given those constraints, bait-fishing for carp is where it's at here, and that's what most people are interested in.
That said, there does seem to be a dedicated community of fly fishers here in Ukraine. Here are few sites I've found today:
Нахлист у Карпатах (Fly Fishing in the Carpathians, blog)
Флайцентр (Flycenter, forum)
Клуб Львівських Нахлистовиків (Lviv Fly Fisher's Club)