It's interesting that the test at 15 only includes reading comprehension and grammar. Unless it has changed recently, the SAT test taken by 14 year olds in the UK must include study of a Shakespeare play.
SATs are taken by children at ages 6/7 (Reading, Writing, Maths), 10/11 (Reading, Writing, Maths & Science) and 13/14 (English, Maths & Science). They are national curriculum examinations which are compulsory within the state sector (some private schools choose to set their own end-year exams).
GCSEs are compulsory examinations taken at ages 15/16 during the final two years of secondary school. They are a two-year course with the exams being taken at the end of the second year. Typically, students will study for between 8 and 12 GCSE examinations - English Language and Math are compulsory subjects, otherwise there is a choice (although some state and most private schools make English Literature and French compulsory, too). Religious Studies and ICTare compulsory at state schools, but not private schools.
After GCSEs students can continue with education by enrolling at a sixth-form or college to complete AS exams, followed by A2 exams (AS+A2=A Level). Students typically study 3 or 4 subjects and have a choice between all major disciplines. A-Levels are the "currency" for most university/degree courses.