You can try to use a compass cutter like this:
I find this product rather effective on materials like leather or felt. The only problem could be if your felt is too thin, in this case it could be too flexible and it could move around while you move the compass, but I have only had this problem when I tried to cut household linen.
That sounds like a perfect use case for needle felting. Instead of using ready-made sheets of felt, use the loose fibers of rowing or batting to create the felted face right on the spot.
With a bit of practise you can achieve colour effects similar to painting by layering and mixing different colours, like the blushing cheeks in your turnips.
In case anyone has a similar need in the future, I'll add a tool. The question doesn't indicate the circle size, but if you're using a plastic stencil, I'll assume the circles aren't very big.
You can get inexpensive sets of hollow punches sold as gasket punches. They're typically a collection resembling this:
Image courtesy Amazon
I bought a cheap ...
Spray adhesive is a good match for this application. If you used the lower strength type, you might even have a decent chance of getting the felt back off later if you needed to for some reason.
Using spray allows you to get an even coat over the entire felt surface, avoiding lumps and/or hard-spots in the finished surface. The hard spots can arise where a ...
I tried an experiment on a square of random tan felt I had sitting around. I chose the letter B because it has a good mix of straight and curved lines, so I could also get an idea of how easy a particular marker was to use.
Clockwise, from top left:
Permament Marker (Sharpie)
Washable Marker (Faber Castell)
Fabric Marker (Marvy)
Fabric Paint (Tulip)
Another option is to needle felt the two together.
You’ll need loose wool fiber (cat hair and dog hair also work great) and felting needles (very sharp, use with caution!).
There’s lots of great tutorials on needle felting on the internet.
Any sort of glue you try to use will be ineffective.
Felting the two pieces together is simple and permanent, if you get ...
Glue is not a good way to add fabrics to fabrics for children to play with.
Glue can get lose and the small parts you have glued on can get in the childs mouth or nose. I have never seen a glue that can not be peeled off by clever little fingers.
Instead of stitching on pieces of felt you can embroider details.
Here is a Felting Needle Chart, via Paradise Fibers, which describes each needle type, gauge, blade style, fiber type, and uses:
Additionally, here is a very helpful YouTube reviewing the types of Felting Needles, demonstrating their various uses, and mentioning safety when Felting.
Embroider or chrochet the face on it, then put the whole thing in the washing machine, so it is felted completely. Though you've already stuffed it, and you probably don't want that to be felted as well. So this is probably an idea if you can remove the filling, or if you're planning to make another one.
If your circle have a limited variety of diameter and the felt is firm enough you can consider the use of a puncher like the ones used in scrapbooking or leather puncher.
I don't have those available currently, but here is a little equivalent experiment with an office puncher: One sees that the punched disk will stick to the main piece on one side (right) ...
Screen printing can be done successfully on felt, but it takes a little bit of experimentation due to the rougher texture of the fabric. Screen printing is also useful for making multiple copies of an image - unsure if you plan to make just one or multiple.