No. Whereas "dy gar di" and "dy gar" are both used in the colloquial language, the "di" is found much less frequently in the formal language. It'd only be used very occasionally, mostly for emphasis or to fit the metre in such things as poetry.
It isn't more nasal in North Wales. The tongue is just slightly further back (and possibly lower) in the mouth. It isn't found in any variety of French that I'm aware of. It's the sound of Romanian î or Russian ы and similar to some varieties of Swedish i.