local government vs regime

local government and regime both are nouns.

Word	Noun	Adjective	Verb	Adverb
local government	Yes	No	No	No
regime	Yes	No	No	No

As nouns, regime is a hypernym of local government; that is, regime is a word with a broader meaning than local government:

local government: the government of a local area
regime: the organization that is the governing authority of a political unit

Other hypernyms of local government include authorities, government.

local government (noun)	regime (noun)
the government of a local area	(medicine) a systematic plan for therapy (often including diet)
	the organization that is the governing authority of a political unit