Which universities offer DLitt via distance learning in its entirety?

Answer 1

None, unfortunately, if you mean doctorates that are similar to the PhD, EdD, etc. I've done an exhaustive search for distance-only DLitt programs in the English-speaking countries but I was unable to find any.

Some Commonwealth universities offer the DLitt as a higher doctorate, but that's a different thing entirely. Higher doctorates are awarded on the basis of published work; they represent the culmination of a distinguished research career. In the natural sciences, a successful applicant would generally have more than 100 peer-reviewed articles in good journals. Higher doctorates are "distance learning" in the sense that the publications are the only requirement -- and they can be completed anywhere -- but I'd guess that fewer than 1% of the faculty at well-regarded universities would quality for a higher doctorate.