Why would you buy new in the UK to ship? Theres some things that are cheaper but on buying loads for a home the overall price would be about the same. Were you thinking of specific things or a whole house? If anything I'd buy clothes to take, cheaper and better quality.
As for shipping stuff you already have it depends on your stuff. Where you bought it, how attached you are to it, how old it is, if you want familiarity around you (great if you have kids but also just nice to have your own stuff) and how much you have to spend in Australia ie if you will need your money to live on!
Stuff from argos and the likes, ditch it wouldn't make the journey and if it did it will be wobbling so bad you will want to ditch it.
Some IKEA stuff is great as you can take it apart and put it back together good as new unlike many other stores stuff.
Some electricals, kitchen stuff etc are cheaper but if yours are not that old may be worth hanging on to them. If older than 5 years may be worth buying new if you would need to replace in the next 2ish years anyway.
It generally works out much cheaper to ship your stuff but thats after taking all of the above into consideration. Get at least 3 quotes to compare the costs with them coming to your home to do an accurate assessment of what you will need.
It was just our friends who live in Brisbane have told us to buy new in the uk and ship it over as it s either rubbish quality or very expensive in Aus. We are taking some of our own stuff over with us but there is a few things like Sofa, Dining table & beds that we need to replace but wasn't sure if it's that much cheaper here in Scotland, plus we'll need a container anyway to ship my hubby's tools over.