Your nationality determines the application deadline, not your residence permit. For example, if you have Mexican nationality but a Dutch residence permit, your application will follow the deadline for Mexican nationals.
Make sure to check your deadline based on your nationality. UT will verify your residence permit to ensure it is valid for your studies. If a new study visa is needed, UT will apply on your behalf, and therefore you must follow the relevant visa application process and deadlines.
During your application, upload a copy of your current residence permit (front and back) when asked for your identity document.