Did a similar trip in 2004, left Perth for Adelaide, stayed with wife's sister for a month and then headed up to Port Augusta and up the Stuart Highway to Darwin.  Came back via the West Coast of WA.

We left Adelaide the last week of May and had great weather with the exception of Alice Springs which had rain of some description each day for the week we were there, late June.  One hours drive North of Alice saw us in some great weather until we got  down to Geraldton where we encountered some much cooler, read cold, weather.

The trip you are planning is great, but you could run into some adverse weather crossing the Nullabor on your way back.

We done the west coast two yrs ago, north to south, late august & early sept was perfect for the wild flowers around Mullewa & south to Coorow,weather was great. Arrived Perth mid sept & it turned cold & wet for six weeks all the way around the bottom & over to SA
Port Smith is very interesting, dont miss the blow holes at Quobba , the Pinicles & New Norcia are well worth visiting.
Enjoy 

As a rule of thumb it's hot and humid up north in the tropics, from October to the end of April.
It's the "dry season" in between, generally.
This year has been the exception. Just ask any southerner who spent time in Cairns this season.
The humidity has averaged 70%, and it's rained on an average of 3 out of 7 days all the way through.
As soon as the humidity climbed a little, the Vic's bolted south again.
Ideally at this time of the year the temps are mid to high 20's in the day time, the sun shines consistently, the breezes are balmy and nights are pleasantly cool for happy hour. Sometimes even cool enough to finish the night with snifter or 2 of port.
The central desert regions have extremely cold nights during the winter months, eg June to August. The frosts are penetrating.
From now on the "build-up" causes the temps to rise with the humidity, and you may see some spectacular, dry thunderstorms.
The westerlies, sometimes stiff, prevail along the west coast and the Nullarbor. The SE trades prevail on the east coast. All of this is subject to change with little notice.
I'm a weather watcher. It's been a lifetime hobby.
A reverse-cycle aircon and a blower heater will see your right in all conditions.

We did a 3   month trip last year from Adelaide up the centre and across to WA and down the west coast and back along the Nullabor and home to Adelaide.  We left in July and found the weather was warm by the time we got to Alice Springs and from there we had almost perfect weather until we got to Fremantle.  We had to stay an extra day there because of the wind and rain - nearly blew us away!!  From there the weather was cooler but still good for travelling but it  gradually got cooler as we got closer to Adelaide.  Hope you have a safe trip - we would love to do it again and see some of the things we missed but it was wonderful and its always good to leave a few places to see for next trip. Last year was a very dry year and we found the waterfalls not as good as expected and some of the campsites like Mary Pool were dry but it was still fantastic and we'll go back again when they have another season like this year.
Ron & RosieB