On both sides of the pond, school as a noun can refer to an institution for educating children (e.g. primary school), a unit or faculty of a university (University of Cambridge School of Technology), or a specialized training institution (e.g. dance school).
On top of this, in North American English, any formal education can be referred to as school. While the pre-kindergarten level is called pre-school or, at early ages, nursery school, American parents might answer yes if asked if their toddler was in school (though they may specify well, preschool, not school school).
School also encompasses undergraduate and postgraduate degree programs. ODO marks this usage as "informal," but I find that to be an overstatement. To pick a few examples from Google searches, Stanford uses the term school year in an official release; the Ph.D. admissions FAQ for Wash U answers whether someone can "keep my current job and go to school at the same time"; but there are also countless official biographies and obituaries which refer to someone's achievements while in school or at school referring to their university and postgraduate years, not to mention idiomatic phrases like working your way through school or school spirit in the language.
Moreover, this usage has been around for a while; while it isn't the easiest thing to search for, I found in the May 11, 1868 Oneida Circular in Google Books
[We] bought a house in New Haven, that we might send others to school at Yale.
And if you object that the line refers to one of the schools at Yale, the 1899 Report of the federal Commissioner of Education refers to an instructor
schooled at Yale, Chautauqua and other summer schools
and if you object that this is a verbal usage, by 1909, the influential humor magazine Puck could offer this synopsis of a life:
The New England Conscience was born in the Back Bay, went to school at Harvard in the early days, lived in Concord, and died in the New Hampshire legislature.
In British English, school is not used to refer to universities or graduate instruction.
Similarly, Canadians and Americans use student for anyone enrolled in an educational institution, whereas on the other side of the pond, it is restricted to university students; those in primary and secondary schools would be pupils. See "Pupil" or "Student", what is the correct use? for coverage of that difference; pupil is far less common than student here.