Hey there! I just graduated with my masters in SLP. With a 4.0 GPA and BA in psychology, I think you would look like a good candidate for grad school admissions but you should know that you will probably have to do about a year of post-bacc work (such as speech science and phonology classes). I believe you will also be required to take the TOEFL (Test of English as a Foreign Language) to demonstrate proficiency.
I did have one ESL classmate (Polish) who got into grad school just fine. None of my professors ever told him the ESL aspect would be a problem. I think as long as your spoken and written communication is proficient for academic standards, you're in especially good shape. Bilingual SLPs are a hot commodity. Do you understand the underlying English grammatical structure well enough to target specific areas? I assume it's different for every program, but one requirement for an undergraduate degree at my school was to take an intense course on English grammar and syntax. You could look into doing that if you're feeling uncertain.
My other suggestion would be to make your recent acquisition of English a positive: you could provide a different kind of perspective when it comes to targeting English language and phonemes compared to native speakers. Being a native speaker myself, I've never had to put thought into the way sounds are produced; you obviously have if you don't have a very detectable accent. That, in my opinion, is something unique you could offer to your patients/students.
I hope that was helpful. Good luck to you! :)