It wasn't clear from the question whether the use of english was a requirement of the class. If so, then it may well be a requirement of your job as well. And if so, then you need to enforce the requirements if you wish to keep the job. Check with your professor about if (and if so precisely how) you should go about doing that, then you are safe from any fallout if the student gets embarrassed or whatever.
Now as for why you shouldn't simply let students flout rules, there are additional downsides the other answers might have missed:
It can be difficult to accurately translate the nuances of a problem or concept back and forth, causing students to be mistaken on some points on the homework/exam. You will likely make real mistakes in translation too. And they will point the finger of blame at you when they complain about their grade to the prof.
It can just generally be quite a bit harder to use two languages at the same time versus just sticking one, even if it's not your first. It takes effort and a little time to change your thinking back and forth. Some students may not be able to keep up, and will miss things as they reorient to the language switches, thinking you said "A=B", when in fact you had said "given C, A=B". Again, blame will be on you when they write this wrong answer on the exam.
Overall, as the tests are certainly in English, students who have not been practicing doing the problems and understanding the concepts completely in english will have a disadvantage. Unless you are there with them for the tests to help them translate every question to be sure they get its point, they will likely miss every question that has a nuance or "trick" to it, and requires careful reading. (not to mention putting downright scary-looking responses on essay questions).