Is it possible to fly internationally understanding only English for radio communications? Are there exceptions?
ATC and pilots must be able to communicate in English on international flights involving ICAO member states. There are 191 countries that are members of the ICAO, covering almost all of the world.
While communications may be performed in the local language, English must be used upon request on international flights. Controllers and flight crew engaged in communications for international flights should be proficient in spoken English in addition to English aviation phraseology.
Some domestic airways and airports may require communications in the local language instead of English.
So English is sufficient if you stick to international airports and airways, but be sure to check the language requirements before you venture into foreign domestic airspace and airports.
ICAO has standardized on English (PDF), which "urges" (apparently it does not require?) contracting states (participating members) to use English for all communication. I believe proficiency standards implementation was originally set for 2008, but was pushed back to 2011.
The exceptions to this standard would be, most likely, any country which is not an ICAO member. For the most part, as of now flight in ICAO member countries should be possible if English is your only language, as the proficiency standards testing has been rolled out.
Personally speaking, I recently flew in Iceland, which has some forbiddingly-hard-to-pronounce names, but is an ICAO member state and which uses English for everything. It was a great, low-stress experience; air traffic controllers were very understanding and knew what I meant when I attempted to pronounce place names, even if I butchered them. Which I almost certainly did.
I can only speak for Europe and the U.S. but be prepared to encounter controllers who only speak French when flying in France. As far as I know, France is the only country in Europe where English is not required for ATC. The same may be true for (former) French colonies but I don't know for sure.