I think it depends on what part of the country or world you come from, and what your parents used growing up. I'm pretty sure it's not an AS or NT issue, judging by both experience and the myriad of responses here...
I always thought "next weekend" was short for "the next weekend after this weekend." Ideally it would be used in a context such as, "Is that this weekend or next weekend?" to make it clear they are two different weekends.
As for the original question, it would be a bit strange to just say "next weekend" on a Monday. Wednesday or later would almost always be "this weekend."
This reminds me about the various meanings of "last night." Around here, "last night" invariably means the last evening I was up before I went to sleep (that is, the day before). In some places, however, I believe this is referred to as "tonight," as in, "how much sleep did you get tonight?"
Interestingly, my wife refuses to advance the calendar day in her head until after she goes to sleep, even if she is up after midnight. So if it's Saturday night and we go to bed at 1 a.m. Sunday morning, we still went to sleep on Saturday night in her opinion.
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"That leap of logic should have broken his legs." - Janissy