In American English, we always say "this time around". In fact, we always use "around", not "round", as an adverb. In British English, I believe that they use both forms. Note that the initial "a" in "around" is a schwa, and in this phonetic context, people will not really notice if you pronounce it or not.

It's not a phrase I use frequently, but "this time around" is the better choice.

Yea, "round" I think is a more poetic/artistic way of speaking. You heat it more in poetry, music video's/songs, and in certain neighborhoods/cultures. But most people, I suppose us "around."

this time around

this time around