across the nation vs nationally

across the nation and nationally both are adverbs.

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

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

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

across the nation (adverb)	nationally (adverb)
extending throughout an entire nation	extending throughout an entire nation
	with regard to a nation taken as a whole