Welcome to the community.
Unfortunately I have to be very honest with you that “Upgrade” is like playing lottery.
Firstly, you will need to purchase the most expensive flexi economy fare, which usually retails well over $4000 return to the US.
Secondly, upgrades with Qantas doesn’t work on first come best dressed, but on your status with the airlines. So unless you are a platinum or gold member, your chances are really small.
Thirdly, it depends on the demand for the flight, ie. dates, season and route. Australia to USA is the one of the world’s most competitive routes, and it is known by expert frequent flyers, point hackers as the world’s most difficult route to find award seats. Dates wise, even though you tried to fly before school holiday, please remember, premium cabin demand doesn’t depend on school holidays but on business activities. School holidays are usually for leisure travelers and they usually compete for economy class (that’s why you usually can’t find cheap economy tickets in school holidays) but parents with kids don’t pay thousands to fly business class, it’s people who need to go for business and paid by the companies. Unfortunately, end of September is the end of fiscal year in the northern hemisphere so I would say the demand for premium cabin is very high.
So if you don’t get upgraded, you are stuck in a very expensive economy class seat for 18 hours.
But to answer your question, the days of the week that will get you a better chance are Tues-Thurs. Season wise? Believe it or not, flying on Xmas and NY days will actually get you a better chance, because 1. most companies are closed for business, 2. Nobody want to spend their Xmas or NY on a plane.
Virgin works slightly different. If there is an award seat available (which is also very rare) you can secure the upgrade at time of purchase. If no availability, then you are in the lottery game as well. But depending on how many Velocity points you have, I would redeem Etihad to New York for 125k points in business class one way. It is the long way round, but you will have a better chance of scoring a free (well, almost free, maybe $100 in tax) business class seat without worrying whether you get upgrades or not. I have flown Etihad’s business class, you should aim for A380 or 787, it’s almost as good as first class. Keith has written a guide on how to redeem Etihad seats using Velocity points.
There are other airlines that can also secure upgrades at time of purchase, these include Singapore Airlines and Cathay Pacific, subject to availability. You can convert your velocity points to Singapore’s Krisflyer points. I don’t recommend upgrades with Singapore either as their miles required for upgrades is almost as expensive as redemption for free ticket outright, and you still need to buy the most expensive economy class ticket. If seats available I would just redeem the seat outright.
Cathay is probably the best for upgrades. The miles needed is much less but you can only upgrade to the next cabin up, eg. from economy you can only upgrade to premium economy. However, as you have to buy the most expensive ticket anyway, you can buy premium economy and upgrade to business, so if you don’t get upgrade, you still have premium economy. Also note that Cathay has been known to release all but 1 unsold seat at the last minute so you have a higher chance of getting upgraded. However, you don’t have any Cathay Pacific’s Asia Miles so it’s not really an option.
There are other ways where you can fly business class cheap. Although not as cheap as flying economy sale fare, it is definitely cheaper than full flexi economy and awaiting for an unlikely upgrades. that is, to purchase airline miles. Keith has just published a guide on purchase airline miles, I highly recommend that you take a look. I personally have a trip to Chicago next year and it costed me USD $2400 return mixed first and business class in ANA and Singapore airlines (New York would be the same cost), not in dodgy airlines with poor service or safety record.
In your case, you could purchase Alaskan miles and fly Cathay Pacific or Lifemiles and fly ANA to NYC.
Lastly, there is nothing to stop you from mix and match with all your points. You might not have enough points in one program to redeem return, but you can do one way SYD-JFK using velocity to fly Etihad, and coming back on Cathay Pacific using Qantas point or purchased point…etc, etc. By looking at one way at a time improves your chance of finding availability.
Hope that helps (and hope I haven’t confused you…). Good luck!