You probably need to get some books on papier mache from your library, or buy Papier Mache Art and Design advertised on this site.
Then really study cats.
DavidO

If you have some oil-based non-hardening clay (often used by kids), you could sculpt the head and neck out of the clay, and when it suits you, start applying paper/adhesive strips or pieces. 

Most of the paper pieces will have to be rather small; larger pieces will wrinkle, smaller pieces will fit together better for a smoother finish.  You will need at least three layers of paper, maybe more, depending on what kind you use. 

If possible, use a colored or tinted paper for the alternating layers, so you can make sure you're making complete layers.  When you use all one color (like white), it quickly gets difficult to see how much you've applied.  Smooth down each piece of paper well, and try to avoid wrinkling, which tends to produce a rougher surface.

Let the head dry thoroughly but don't use heat.  Then take a sharp craft knife and slit the head from the middle of the back of the neck, through the center of the top of the head and down the center of the face, and under the chin, so you're splitting the head into two halves.  Remove from the clay form.

As much as you can, use paper masking tape to join the two halves back together, taping the INSIDE.  Once it's together, apply another layer of paper and adhesive on the outside to conceal the joint.  If the form doesn't seem strong enough to apply much pressure, see if you can temporarily stuff the head with cotton or polyester stuffing to give it support.

Cut ears to fit, but since they are so delicate, they should probably be the last part applied before applying your final surface treatment.

For the body, the easiest way to build it might be to start with a fairly stiff wire armature, wrapped and secured with finer wire.  Be sure to include the tail within the total armature for strength.  Depending on your plan, you may want to extend the armature out of the neck so you can attach the head.  Whatever seems best to you.

Then 'bulk up' the armature with small wads or ropes of gluey paper, small wads or 'snakes' of paper pulp containing adhesive, paper rope soaked in adhesive (with excess squeezed out), or  crumpled aluminum foil.  Fill in gaps with small pinches of paper pulp/adhesive. 

Once the body is looking right, start smoothing the surface with either a thin layer of paper pulp, or paper strips and pieces.  Smooth it all down carefully.  Keep looking at it from all angles.

Since you have the original wooden cat, you could lay it on its side on a piece of paper, and draw the outline as a guide.  This can help you determine how long the wire pieces should be, and fit the armature so you don't have ends sticking out (except maybe the neck part for the head).  In most cases, the armature shouldn't be any closer than 1/4" (5-6mm) to where the 'skin' will be.

Attach the head at a lifelike angle and apply more PM (probably strips are best here) to connect the head to the body.

The more papier mache work you do, the more you will understand what you need to do.  Don't expect perfection in the early days, even Michelangelo probably started out with stick people and lollypop trees.

Sue

If anyone is instantly good at something new, I've never heard about it.  Brain surgery takes practice, farming takes practice, driving a car takes practice, so why should PM be any different?

At least PM can be burned or recycled into something else if it doesn't turn out.  I also work in concrete, and the only answer to mistakes there is a hole in the ground or the county landfill!

Sue