nationally vs across the nation

nationally and across the nation both are adverbs.

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

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

Other synonyms of nationally include across the country, nationwide.

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