infant school vs primary school

infant school and primary school both are nouns.

Word	Noun	Adjective	Verb	Adverb
infant school	Yes	No	No	No
primary school	Yes	No	No	No

As nouns, primary school is a hypernym of infant school; that is, primary school is a word with a broader meaning than infant school:

infant school: British school for children aged 5-7
primary school: a school for young children; usually the first 6 or 8 grades

Other hypernyms of infant school include elementary school, grade school, grammar school.

infant school (noun)	primary school (noun)
British school for children aged 5-7	a school for young children; usually the first 6 or 8 grades