The mathematical topics that will be covered in a placement test for Harvard University are on the Harvard website (link attached below). They include fractions, algebra, functions, graphing, differential calculus, series, and differential equations.
British universities such as Oxford and Cambridge do not use placement tests. This is because students are admitted to study a particular subject and do not then have much choice in the course of study. Even admissions tests have become fairly rare, as students are usually admitted based on their final examination results at secondary school. However, it is possible to take standard admissions tests in mathematics at Oxford. Here, the test for entry includes polynomials, algebra, differentiation, integration, graphs and transformations, logarithms, geometry, trigonometry, sequences, and series. A fuller explanation of the syllabus is attached below.
Both Oxford and Cambridge Universities are made up of various constituent colleges. To gain admission to the university, a student must apply to the college. Mathematics at Cambridge is highly college-based, and it is the college's choice whether the student takes any type of entry test and what is covered. In any case, the test will not be a placement test. You may have heard of the TMUA (Test of Mathematics for University Admission), which is administered by Cambridge University. This is accepted by several British universities, including Durham, Lancaster, Sheffield, and Warwick but, ironically, not Cambridge.