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.
This will help as is more thicker than ordinary glue
You're probably more familiar with this one, the pros with this one are that it adheres and dries fast.
Fabric and Permanent glue
proceed with caution with these...
The strongest hold but, there is a con here: the smell, but it's not that kid-friendly.