You could use velcro with an adhesive backing, but with a foam rubber nerf ball, I don't know how well the velcro's adhesive glue would hold. At least if you tried it you can always reposition it.
Sewing is out because the Nerf ball is bare foam rubber and it would tear.
You can glue the nose on with a hot glue gun. But hot glue can make parts hard and stiff when it cools down. But it depends how much glue you apply.
There is also contact cement, but the strong fumes the glue gives off are best to use outdoors.
Crazy Glue can be a good choice too if used in small dabs.
But with any glue, once applied, the original material will never be the same if you try to remove it.
If you are not sure of the final look for now, you can always pin it.
Many 'what not' Muppets have their noses held on with pins.
If the puppet head and the nose were eventually covered in fabric. I would sew it.
Anytime you can avoid glues, your healthier for it.