In the States, an elementary school and a grammar school are the same thing. But a search revealed that an English grammar school involves students from 11 to 18 (according to the source I found), which is quite different.
In fact the term "elementary school" is no longer used in England and hasn't been since at least the end of world war two.
In the fifties and afterwards there were
primary schools (a term still used today) which could be sub-divided into infants and juniors. Primary schools were for childern from 5 to 11.
At age eleven you took a very important exam called the "eleven plus".
This could open the doors to a
grammar school. If you were less successful you went to a
secondary modern school. The former were single sex and the latter were for both sexes.
John, Paul and George were intelligent kids and earned places at grarmmar schools. Ringo failed his eleven plus and went to Dingle Vale Secondary Modern School.