Someone can correct me if I'm wrong but this is how I understand it.
Near and Nearly
Source: Grammarly Handbook
Nearly is an adverb, which means almost; near is a flexible little word which can be an adjective, verb, adverb or preposition.
This means that both of the following are correct:
This essay is nearly complete.
This essay is near-complete.
If we switch to "Nearly universally", we are now dealing with "near" + adjective + noun.
An example sentence could be:
This essay is nearly universally accepted.
This essay is near-universally accepted.
Here, some minor ambiguity is possible because there are two modifiers and a noun. To me, these both can have different meanings, if only slightly.
This essay is nearly universally accepted. The essay is very close to being "universally accepted".
This essay is near-universally accepted. The essay is accepted "almost universally".
Nearly modifies accepted while Near- modifies universal.
Neither of them sound great to me and neither is widely used. But of the two, nearly universally is more common according to Google NGrams.
I recommend skipping both and rephrasing into something cleaner.