Can’t say if it’s realistic or not but I would suggest finding a running coach and paying for a few months of coaching. A coach can help you with making sure you are training correctly, making certain you aren’t adding miles too quickly, supervising speed workouts, etc. They will also help you avoid injuries. Well worth the money.
I am not sure why you think being able to run one 6 minute mile will get you on the cross-country team. At the very least (assuming you are a girl) you will need to put three of those back to back with uneven footing, hills, non-ideal weather, etc. If women at your school are running ~ 18 min for a 5K, it, means that on the track, they are well under 6:00 for a one mile time trial. Even if you can get to that point, the coach will also want to know why you did not run track in hs. Competing at the college level is a big commitment, and in order for the coach to give you a roster spot you will need to show that you are ready. If you competed in hs and are just out of shape looking to get back into it, that is different, but from the way your question is worded it looks like you have just now decided to give running a try and are completely under calling what it will take to be competitive.