What would be the best way to to go about making this recipe dairy-free?

There are many good non-dairy/vegan butters, milks and cheeses available, which would be easy substitutes. Melt butter is my favorite non-dairy one. I like oat or macadamia milk. Violife and Miyoko's have good cheeses, among others. I prefer these to nutrient yeast.

You could use oil (your choice) at 80% volume to the butter measure. Or, as you suggest, a dairy-free spread at 1:1.
For the cheese, you could just omit it, or as you suggest, use nutritional yeast. I'm not sure of the measures so I would start with a small amount the first time you bake these and add more (in the baking or as a sprinkled garnish) if you feel the muffins need it.