As far as I know, there are no direct Krisflyer miles earning cards in Australia. Has to be transferred from Velocity points (in a bit of a limbo at the moment) or via flexible points like Amex pts.
If you have a KF earning card in Singapore and it is financially feasible to keep using it (including forex costs, etc), Singapore issued cards may have a superior earn rate. In Australia, max earn rate for Amex is ~ 1 KF mile per dollar on Amex Explorer or 1.125 KF mile per dollar on Amex Platinum Charge (which does come with a heftier annual fee).
Another thing to consider is the partner airlines that Krisflyer or Qantas have access to. Krisflyer, naturally you can catch Singapore Airlines to EU. Krisflyer members generally have better award availability than partner airlines (e.g. Star Alliance airlines). For Qantas, you would have limited routes on Qantas metal (to UK), otherwise, you could take Cathay Pacific via HKG, Emirates via Dubai, etc.
Krisflyer redemptions in general have much lower taxes and surcharge when compared to Qantas.
However, there are endless more ways to earn Qantas points in Australia. So it is a bit of a mixed bag which way you go. If you are going to do lots of domestic flights in Australia, you would generally be earning Qantas or Velocity points (when they resume as Velocity 2.0). So that may be a factor as to what points you can generate faster to a useful balance.
If you can generate lots of points via credit card expenses, it is generally better to earn both (more) type of points in the long run as award availabilities and routes varies between program. However, there is bit of a period or transition point at the start of your point hacking journey, where it is best to focus all your efforts to generate enough points in one location to be useful for your needs (e.g. enough for return flights for one or two pax).
Good luck with the move.