Anyone good with Danish fabric ID?  

Ha, that's the sofa in my friends' shop!  I was just talking to someone this past weekend about the cushions on it.  The fabric is something I have not seen on anything else and it's not one of the ones that Kvadrat still makes from that era, but it's definitely a natural fiber.  Maybe a wool/rayon blend, I'm not sure.  I can look at it more closely next weekend if you like.  It's not a pure linen or cotton, I know that.  And I'm quite sure it's not a synthetic.
I think you can go with any of the nice wools from Kvadrat (Danish company).  Check Modern-Fabrics.com, they often have Kvadrat fabrics listed.  You can figure out the yardage pretty easiy--two panels will fit in one width (54") of fabric very easily and you can get the narrow side pieces out of the leftover on sides. 
I recommend the firmest possible 2" thick foam for the seat and medium density for the back.  Don't skimp on the foam or the quality of the upholstery!  
The cushions on the one above have a nice seam detail which you can just see in that photo.  I'd do replacements like that unless you have some objection to it aesthetically.  I would think that the designer specified it.

OK, what are the odds of that? Baltimore? I have no trouble doing any kind of wood repair work, but fabrics and upholstery is scary for me. Especially since I have to order something then wait weeks before I find out my measurements are wrong. I would love to know the sizes of the cushions - the middles ones are narrower than the others. Yelp reviews say the owners are really nice - do you think they would mind if I called and asked them the measurements?
I noticed the subtle seam, and actually like the look but that's too custom and beyond anything I think I could order.
Thank you for the link to the fabric store - amazing selection and prices! I'll definitely start there.
Do you have a recommendation for a place that does good custom upholstery? I found cushionsource.com and they seem good but they don't give much choice for foam, and I wondered if 2" would be thick enough? They have a 2" cushion but only one density and they call it a chair pad, not a cushion. There are several of these sofas/chairs on the net, including some on this board, that have thicker replacement cushions, some like the link below also on 1stdibs look to be thicker than 5". I was considering going to 3", though I know the look would be different. What do you think - does that ruin it?

Why order finished cushions online?  Isn't there an upholseter near you?  You don't need to find someone who specializes in Danish furniture, just someone who can copy the look of the originals.  They are a very simple style, not really any more complicated than a cushion found on a conventional sofa.  
I think thicker cushions change the whole look of this sofa.  I'd really try to find someone who can do extra firm density 2" thick foam.  I would bet that place you're looking at uses the foam that comes in huge rolls, which is maybe medium density at best.  It's fine for back cushions but in the 2" thickness it's too soft for seat cushions.  I have to buy 8' long slabs if I want densities other than medium.  You could probably find it online and just get it yourself.  There used to be a guy on Ebay who sold all different densities and thicknesses of foam.  
Ordered finished cushions online just sounds like a crapshoot to me.  The cushions are part of the original design and need to have a specific look, y'know?

Just FYI, the cushions in the photos of the sofa in the first photo look a bit thin compared to photos of vintage advertisements. See here for example:
http://www.danish-modern.co.uk/kai-kristiansen-for-magnus-olesen/

And as spanky said, try to get replacements that look the same as the originals. Kai Kristiansen designed it that way. 

Thank you both for all the great advice and links. Really VERY helpful. My conclusion is that they may have made the cushions a bit deeper over time; the earlier cushions from the 50's 2" and later ones in the 60's about 3". Maybe catering to American butts (which could be why aftermarket replacements are getting thicker and thicker ...)
Modern-fabrics has a sale that ends today (a few hours from now), but I'm a little unsure of how much material I need at this point. Do they only have one sale a year, or if I wait for a few weeks will they be likely to have another sale?

Thanks for posting that, Leif--I do think the ones that my friends have started out thin though they may have gotten a bit compressed with age.  But not a whole inch!  The side panels aren't that high.
Gropius, Modern Fabrics has sales every few months (?) but keep in mind that their regular prices are below wholesale on a lot of the fabrics.  Yes--below wholesale.  Not a typo.
I explained how to measure for yardage in my post above.

On danish-modern.co-uk are a few photos of this model and other Magnus Olesen pieces spanning 1956 to 1965, and none show the thin cushions:
http://www.danish-modern.co.uk/magnus-olesen/
Also, this isn't an incredibly rare sofa, so there is a good selection of others to find on the internet.  I didn't see any thin cushions in Google.
Not conclusive, but suggestive.
As a contra argument, I will mention that the only situation where I've seen such thin cushions is over experspring.  (E.g. Juhl's Spadestolen, or Hans Olsen for Juul Kristensen rocking chair model 532A).  And this sofa does have an experspring foundation.  So...?
 

I dunno.  Maybe the thin ones are replacements made long ago?  Or maybe they're a rare prototype, making the sofa worth $47,000.  (I kid, researchers from the future.  I kid.) 

I have ordered replacement foam cushions from several places, and I liked this small family business in Minnesota the best. You can select your preferred density (they'll help you choose) and they cut it to size. It's called Best Foam Cushions dot com. It's not the very cheapest, but great service and they're unbelievably friendly. 

Mine are in pieces since they need some wood repairs, and I didn't want to judge the cushion sizes without having them in their final state. Based on how easily they came apart (except for some idiot who did repairs with finish nails and screwed on plywood) I would bet these were shipped from Denmark flat, then glued and assembled when they arrived.
Do we know for sure that pieces like this were shipped with cushions, and that the cushions weren't added in the country they arrived in? That could explain some of the cushion variation (although of course could also be replacements over time). Given the desire to ship flat to save shipping expense, I could see the benefit of saving on the cost shipping bulky cushions too.
Also, did anyone notice that there are two different styles of these (besides the varying cushion sizes)? The first pic in this thread shows what I think is the older style, with a horizontal and fixed width side stretcher. The second pic shows a slanted and widening side stretcher. The danish-modern site shows the (older) one only in an ad from 1961. The slanted stretcher is in ads from 62, 64, 64 and 65 and catalogue from 59. Maybe it took some time for the newer design to get from the catalogue to the ad copy?

If the manufacturer did look to minimize the size for international  shipping, (seems plausible that this happened in some cases), wouldn't it have made the most sense to ship without foam, but with the empty cushions?

Gropius, I looked at my friends' sofa cushions and now I am not so sure that they're original.  All of them have a strip of velcro sewn across the back but of course there is no corresponding velcro on the seat or back of the frame.  So they must have come from something else, though they do fit perfectly.
To find the width for the cushions, just measure the frame between the arms and divide by three.  The one here has a narrower center cushion (two 21" wides and one 18" but don't go by those numbers for yours).  The back cushions on this one extend about an inch past the end posts of the frame.  The actual height measurement will depend on the thickness of your seat cushions, since the back cushions rest on the seat cushions.