How do you get a mugo pine to backbud? I have a beautiful trunk, but the branches and needles are a distance from the tree. First time with a pine.

My method ( works in my climate, on my trees, so dont take my advice as absolute truth)
Make sure your tree is vigorous first, don't do this on a weak tree, or recovering from a repot or heavy pruning.
In spring, when buds have opened, but needles ar not hard yet, remove all of them simultaneously. I get backbudding even on very old wood.
Cut back the next year, let the tree recover, repeat

My experience is that they backbud quite good if one decandles in spring.

Forgot to say I am in zone 6a/b in the USA

I am in similar to zone 7 in Sweden. My trees are protected in unheated greenhouse during the winter as we dont get much snow. I think that the longer growing season I get is beneficial for the backbudding.

Mine is currently in a heated greenhouse. Heater comes on when the temperature is less than 35 degrees F. Plan on taking it out middle of March.

Sounds like the pine will have a good start with you. Just be careful when sun heats the greenhouse in spring. Ideally you dont want the temperature to raise above 10 c in the days before you move it outside in spring.

When the outside temperature is above 35 degrees f, I open the door and roof window to keep temps lower. I have a thermometer is the green house that tells inside and outside temps. When it is above 40 degrees f in the greenhouse and outside, it gets opened up.