Happily, he helped me out.
運良く is more directly 'fortunately' and often used in slightly more formal contexts, while うまい具合に has a casual, conversational feel.
たまたま means 'by chance' or 'coincidentally', without the positive nuance of a lucky outcome that うまい具合に carries.
Standard kanji-kana mix; the most common written form.
Alternate kanji for うまい; less common but still encountered.
Phrase combining うまい (good, skillful) and 具合 (condition, state) with the adverbial particle に. The exact historical derivation is uncertain, but it functions as a set adverbial phrase meaning 'in a good way' or 'luckily'.