I don't know much about antiques, so please understand that these are just some thoughts.

When furniture was being made of papier mache, they used heat and pressure to laminate the paper together.  And I think some of the paper included plant fibers (cotton, flax, etc), but I don't know what percentage.  Wood-based paper usually has short fibers, high lignin and high acidity. Today, we mostly just use adhesives.  Melding the two could be tricky.  If this antique is not especially valuable, maybe it wouldn't matter what paper you use.  If it is valuable, you might be able to pick up some valuable tips if you contacted a museum or two.

You didn't mention how thick the missing piece is.  If it is very thick (5mm), and you are using thin paper, it could be a very time-consuming operation.  A fiber-based paper might be thicker and require fewer layers.

Whichever you use, you will probably need to use a strong, rigid, flat piece of wood (like plywood), cut to about the same shape as the patch will be or slightly larger, and a clamp (a C-clamp would probably do).  You will also need silicon-treated baking parchment or a small piece of sheet teflon (also used for baking).  If you attempt to clamp the piece of wood to the glued paper, it would probably all stick together, creating more of a problem than you've got.  If you use baking parchment, tear off a test piece and submerge it in a glass of water for several hours.  The right kind will not absorb any moisture.

Make sure your wallpaper adhesive is the methyl cellulose (MC) of the CLEAR type.  White or cloudy types may be food-based, or simply be refined flour adhesive (some wallpaper adhesives are just flour).  Plan on using just thin applications of the adhesive. Thick gloppy applications are not appropriate.

Clean the area of the patch well, and remove any coatings or glues.  If possible, sand it smooth.  To keep it as simple as possible, you might want to cut with a small, fine-toothed saw, or sand any rough edges where your paper will be fitting into, or you will have to hand-fit each piece of paper individually into the space.

Tear your paper and see which way the grain goes (it will tear most easily with the grain).  Plan on alternating the grain direction for each layer of paper for strength.

I would tend to think you should only coat one side of the paper with your adhesive, but that is just a feeling.  Put it in place as smoothly as you can, and press it down firmly, squeezing out any air bubbles.  If you have one of those small plastic squeegees for laying down vinyl or smoothing ceramics, it might be helpful. [http://www.bigceramicstore.com/supplies/Sieves.htm
See the ninth photo down on the left to see this item -- it's blue.  Click on the photo for a better look.]

You could probably apply two to four layers of paper, depending on it's thickness.  Top with your handcut piece of parchment or teflon, put your handshaped piece of flat wood on top of that, and clamp it all.

Using methyl cellulose, some of the moisture will be absorbed by the chair itself.  Leave it clamped for a day or so, then remove it and apply another few layers of paper and repeat.

I would think you would want to build up the layers until they are slightly higher than you will want, so you will have some room to sand the patch down to the height of the surrounded material.  Make sure the patch is THOROUGHLY DRY before you start sanding or applying any surface treatment.  Mold from incomplete drying is your enemy.

When you are applying your surface decoration, apply light, thin coats.  Heavy wet layers can damage the surface of papier mache.

And as I said, all of this is theory.  If it feels wrong to you, find some expert assistance.

Good luck!

Sue