This is an odd one. Short answer: No, not really offensive here. But that is changing.

More complicated answer:
We've been learning a bit about how hurtful some of this stuff is as people of African descent come here and teach us. Oddly, the first time I saw a reaction to golliwogs in the media here it was when Oprah Winfrey was visiting Melbourne about 15 years ago and saw them just on display at some store. She obviously was offended by their presence, but culturally, we had no idea why. There may be other factors (Influences of the Internet etc), and they were super old-fashioned before even then. They've really fallen out of favour since Oprah came to town. I'm sure she's fine with that.

This wasn't long after Harry Connick Jr's freak-out on the Hey Hey its Saturday revival special. That was when I learned what blackface meant. The episode made major news in Australia for a good few days as culturally, we had zero idea that blackface was 1: a thing, and 2: super offensive to African Americans. The story there was some Indian dudes had painted their faces black for some skit on the show and Harry would have no part of the segment.

We're isolated down here and just not aware that some tropes are not on. Through Reddit, I learned that apparently fried chicken is another thing that can be considered racist when combined with black people. I'm not really sure how that one works, but there was a KFC cricket ad where someone had KFC in the stands against the Windies and that was somehow offensive as well.