What to use for sofa webbing?  

Those channels are for...
Those channels are for rubber or vinyl covered coil springs (see pic below).....Don't think you will have any luck finding them as they need to be that specific size and, as far as I know, are no longer made

I guess steel wire is good...
I guess steel wire is good for webbing of this sofa. Elastic wire is also good but it will not last longer. And you?ll not find any coil springs that fit for your sofa anywhere, unfortunately.

My understanding was that...
My understanding was that Simon sold replacement material - known as Experspring.
http://www.danish-homestore.com/acatalog/EXPERSPRING.html

That longitudinal wood center support has got to go.
The tubular metal cross-support(s) may be original.
A good upholsterer should be able to either retrofit Fagas straps or build a frame for traditional-style jute webbing that will sit flush to the visible frame members.
Either way, your sofa will be much more comfortable, if not completely original.
Edit: Ah, I see now that you've removed the offending parts! Best of luck!

.
Saddlers have great stuff, D buckles + webbing?

EXPERSPRING
TCHP is right.
I stock the experspring needed for your sofa and they are so easy to fit.
Measure the gap across from loop to loop and multiply by 2 and the number of gaps in the seat.
You deduct 5% in length for the right tension.
Screw the thread half way into one end.
Counter coil the others 5-6 times.
Join the end to the thread and twist toggther.
The spring will then uncoil itself and sit neatly in the seat groove.
Job done for 20 years!!
I sell it by the meter and each meter comes with a connector thread. Each meter is £5 excluding postage.
I would estimate that you will need in the region is 16-18 meters for a complete sofa seat.
http://www.danish-homestore.com/acatalog/EXPERSPRING.html

Thank you all!
Thank you all very much for the tips! I didn't know the coil springs and I'm looking into the Experspring solution. I'm currently still sanding down all the parts though.