Check out Harvey Norman and the Good Guys as they sell TV's, washing machines etc. You'll get an idea of the cost to buy new ones. For our shipping we were only bringing around half a container and paying for volume. So by the time we did the maths on buying new units versus paying for, and waiting X months for, old stuff we decided to not bring white goods. Your maths/situation may differ. Also throw in that should your white goods need repair/service you may find it difficult to get that done. And if anything breaks you may find warranty claims a challenge.
That said, your washer and TV should work fine with a change of mains plug (there's some fluff written about only licensed electricians being allowed to change plugs, I'm not sure how true that is, I could never find a clear answer). But if you're bringing anything with a UK plug, also bring some of those 4-way strips (then you just have to change one plug et voila, 4 UK-plugged items have power). Your TV will likely have a setting in its setup menus to switch it to Australia. But unless it's a very old TV it'll pick up channels just fine. If you'll be plugging it in to Foxtel or some other Set Top Box that may not be a concern
Our mover was very clear that we had to be meticulous with bikes and their tyres. Like, make sure there is not a single spot of dirt anywhere. So we were. No hassle with stuff getting through (other than how long it took).