For now, don't worry about it. So many women have low lying placentas at your stage and the vast majority do what they're supposed to do and move upwards. Having a previa is very rare.
But, since you asked what to expect if it doesn't move up, I'll tell you my story.
They noticed my placenta was low, but I was told basically not to worry about it as they generally move up, so I went about my normal business and didn't give it another thought.
Until I was about 31 weeks along and I got up one morning, walked down the stairs, and when I got to the bottom, I felt like I had leaked on myself.
I went back upstairs, thinking I had peed myself and headed to the bathroom to clean up, and that's when I saw it wasn't urine, but bright red blood.
Panic set in, I woke my husband, an ambulance was called, there was a flurry of activity, and I ended up at the hospital. The bleeding had stopped before I got to the hospital.
They did an ultrasound and saw that the placenta was partially obstructing the cervix. My doctor decided that I would stay in the hospital on bed rest for the duration of my pregnancy because 1) I had already had a bleed and he wasn't taking any chances and 2) I lived so far from the hospital (an hour away) that he was scared if I had a major bleed there wouldn't be time for me to get to the hospital.
We scheduled my c-section (you can't have a vaginal birth with a previa) for 37 weeks.
At 36 weeks, I started having mild contractions (which are a no-no with a previa), so they did my c-section that day.
Everything worked out fine in the end. The baby was 5lbs 10 oz, needed a bit of oxygen for a few minutes, but otherwise fine. That was a Monday, we went home on a Friday.
Basically, if you do end up with complications, you'll have bed rest.
Best of luck to you and your babe.