across the nation vs nationwide

across the nation is not an adjective while nationwide is an adjective.

Word	Noun	Adjective	Verb	Adverb
across the nation	No	No	No	Yes
nationwide	No	Yes	No	Yes

As adverbs, across the nation and nationwide are synonyms defined as:

Other synonyms of across the nation include across the country, nationally.

across the nation (adjective)	nationwide (adjective)
	occurring or extending throughout a country or nation

across the nation (adverb)	nationwide (adverb)
extending throughout an entire nation	extending throughout an entire nation