We moved overseas but I think it's all similar. We got three quotes based on my estimate of cubic meters. I phoned and asked about estimating volume though, people were really helpful. I was pretty close in the end.

The big difference in price for us was the hourly rate, the volume rates were all similar but some charged more for their time than others.
We got the quotes one week and then they came the next. Not sure if we got lucky but we didn't have to wait. But maybe give yourself a couple if weeks to be safe! The truck didn't leave our place and go to Brisbane. It left our place went to the depot in Melbourne and then went to brissy before being shipped. So that might also make a difference to how quickly they can do it iykwim? If they need to schedule the interstate drive ob the same day they might need more notice. I'm only guessing though!!