Not the original poster, but I also started learning a language at SOAS as a beginner last year. The amount of progress you will make in a language depends on what course you are taking. Generally speaking, if you are doing an intensive course, the aim is to get you to the C1 level as per CERF guidelines at the end of your course. Depending on whether you are UG or PG, it may involve a semester or a year or summer abroad, where you will have intensive sessions in your chosen language.
In non-intensive courses, the progress is still quite good, but again, depends on the courses you take and the individual output as well.
-A (MA Iranian Studies and Intensive Persian)