What are my options when trying to adhere felt to a 3D crocheted surface? I was playing OverWatch and saw these guys in an arcade box. 



I started making the green and white parts. What my wife now calls the cute turnip. I am going to use felt for the facial features. Here is what I have so far:



I was considering a large swatch of white for the face and affix the black eyes and mouth on top of that. The white would take up a large contoured surface. Even if I get talked out of doing it this way it would be nice to know what my options are. Gluing comes to mind but there is not a good gluing surface behind it for a good bond. Sewing has a similar issue with not enough to sew into. Ideally I would like to hid how it is attached. 

Help me make my silly turnip thing!
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.

Instructions abound on the net. Felt Alive's seems pretty exhaustive, covering many aspects and with ambitious projects.

LilBlueBoo.com shows how to make a face on a base and Craftsy.com has an image on a knit base, similar to your project: