I would suggest getting an old illustrated hardcopy (not Disney products) and looking at the colours for inspiration for the walls and woodwork. You can match colours and find something that suits -- look online but for a better match go and see samples for yourself. There are some lovely hazy blues and deep creams and even tans in the 'decorations' of E.H. Shepard.
Yellow is a very hard colour to match as it responds to different light in different ways. Bring samples back to the bedroom you are using and look at how it all looks in the light you have, preferably with the curtains in situ. If you choose yellow gingham curtains I would be inclined to choose a different wall colour. The yellow curtains would need to have a muted tone imo -- at any rate not an in-your-face bright yellow. Some gingham fabrics here, on eBay. To give you more choice and more chance of getting the shade right, I would look for fabric and not ready-made curtains, and either make the curtains yourself or have them made.
After that, you are looking for Edwardian-style plain, sturdy furniture, which can be found online or at sales. You can comb through the illustrations to see the sort of furniture that would suit. Maybe a close look at Mary Poppins sets would also inspire you?
Pooh has a kitchen dresser. Iirc, his hunny is kept in a wall-mounted corner cupboard. Both of those items might be found in pine kitchen stockists and you could finish them any way you wanted. There are plain, sturdy wooden chairs and a stool or two, and Wol (maybe someone else) has a chintz covered armchair. There are rag rugs/woven rugs (not tufted), little wooden boats on the mantelpiece, and an old clock or two in the illustrations. I noticed some framed butterfly collections (? - need my glasses) in the link you posted. This sort of stuff can sometimes be found at boot sales or on eBay. Or you can buy kits for things like the wooden boat and paint them yourself.
The trick would be not to clutter but to use decorative items sparingly to convey the feel of the era and the theme of Winnie the Pooh -- Edwardian nurseries were plain places where children consumed bread and milk before bed, and led orderly lives. I think the bedroom in the link is very nice and certainly the right accessories were used, but it is too cluttered.
I like the lack of motifs too and I think such a room would grow with your boys.