The correct spelling for the word "budgy" is actually "budgie", with an "ie" at the end. The IPA phonetic transcription for this word is /ˈbʌdʒi/. The "b" is pronounced as usual, while the "u" is pronounced with the same sound as the "u" in "but". The "d" is pronounced as in "dog" and the "g" as in "go". Finally, the "ie" is pronounced with a long "e" sound, like in "see". So, the correct pronunciation of "budgie" is "BUD-jee".
The word "budgy" is a colloquial term used in some dialects of English to refer to a budgerigar, which is a small parakeet native to Australia. The term "budgerigar" itself has an interesting etymology and comes from the Gamilaraay Aboriginal language spoken in Australia. In Gamilaraay, "gijirrigaa" means "good food" or "good drink". European settlers in Australia adapted this term to "budgerigar" or "budgery-gah", which eventually became shortened to "budgy" in colloquial usage.