I suspect oak, ash, or some type of cedar.
I believe it will always have been ash.
European ash, Fraxinus excelsior, is one of the traditional coachbuilders' woods, valued for its strength and shock-absorbing properties as well as its ability to take bends well.
It's still used today for major structural elements in Morgans:
From what I understand European hardwoods can be very expensive in Australia so it may not be feasible for you to use this even if you can locate a local supplier. American versions of ash could be equally suitable but no idea if they're available there or what they cost.
I'm in Australia - Tasmanian Oak, Meranti, Pine, Oregan are easy to get.
Tasmanian oak (actually some types of eucalyptus) and meranti (Philippine mahogany) are both viable options. Of the two I'd probably favour the tassie oak, in part because it's more likely to be from a managed supply. And this PDF from Forests NSW tells us that it bends very well, The Bending of Timber
Pine I'd discount on principle, regardless of species.... but especially if it's radiata.
Oregan (oregon?) may be douglas fir as one of the alternate names for doug fir is Oregon pine. If that's what it is even though it's a softwood it actually wouldn't be a terrible substitute, it's widely bent e.g. for boats and is stiff and (relatively) hard.
Air-dried v kiln-dried
Although kiln-dried wood is widely noted for being less suited to bending1 (up to and including the opinion "Don't do it.") it's widely used for the purpose as you might imagine because kiln-dried wood is now the norm, not the exception.
The following video on YouTube from Engels Coach Shop, Steam bending wood, 1" thick kiln dried ash. has much useful information. You may also want to have a look at this page on the Cornwall Austin Seven Club's site. A Google search just for "bending ash" will net much more information.
Now, I can't easily get green timber
You wouldn't want to use green wood for this because after bending you'd have to contend with lots of unexpected movement (including some likely warping). In addition, you'd have to wait for it to dry — a year or more! — before you could safely attach it to the existing components and during which time you could expect some cracks to form, even if steps were taken to seal the end grain well.
Expect some learning curve with bending, it's sure to be one of those things where it's more difficult in practice than it appears. There may be some subtle points that you only realise once you get hands-on with the process (there usually are) no matter how much you've read up on it and how many videos you've watched of it being done.
1 One of the basic issues with kiln-dried wood is that it's drier, but the EMC of either wood fundamentally depends on local conditions. In a humid climate wood will bend more readily than in a dry climate, regardless of how the wood was initially dried.