As you suggest, British English tends to just go by year number for University level students:
- Freshmen - 1st year student or 1st year undergrad
- Sophomore - 2nd year student or 2nd year undergrad
And so on until the final year (3rd year for Bachelor's Degree students and 4th year for Master's Degree students), the students of which are referred to as final year students.
Students who have completed their degree and are attempting to achieve a PhD typically do not use year numbering at all, and are merely referred to as post-graduates, which contrasts with students who complete their degree but do not go on to further education and are normally referred to as graduates.
British English never uses the terms Freshman, Sophomore, Junior, Senior etc (in any educational context). Use of those terms is not well understood in Britain, and most British English speakers would not understand what the term means.