I'm an American who has lived in Australia for a little over a decade, and I have not noticed a difference between Australians and Americans in this regard. If you aren't too familiar with one accent or the other, it may be a little difficult to pick up on whether someone is emphasizing or deemphasizing "too" or "bad", but the essential meaning of the emphasis is the same.
In general, I find Americans and Australians deemphasize positive things at about the same rate and in the same circumstances, though the exact phrasing used can differ (although "not too bad" is a case where the phrase is shared in both dialects).
However, Australians are more likely to also deemphasize negative things (e.g., "This weather is average" to mean it's bad) whereas Americans are more likely to use sarcasm for negative things (e.g., "This weather is great" to mean it's bad).
In general, I find Americans and Australians deemphasize positive things at about the same rate and in the same circumstances, though the exact phrasing used can differ (although "not too bad" is a case where the phrase is shared in both dialects).
However, Australians are more likely to also deemphasize negative things (e.g., "This weather is average" to mean it's bad) whereas Americans are more likely to use sarcasm for negative things (e.g., "This weather is great" to mean it's bad).