Yes phoning them up could work. However, most sixth forms will have a formal process e.g. you may have to attend a specific day with your GCSE results to enrol into that school for the subjects you want to study.
I imagine when you phone up, they may tell you that you have missed enrolment day - but some ofc may be willing to accommodate given your situation. The unfortunate case may be that even if they accept you, they may not have any spaces left in the subjects you want to take, and worst case is they don't admit you at all due to passing of enrolment deadline.
Thus, it's best to research local sixth forms that you want to go to near your new place and reading up on their sixth-form enrolment procedures. You can call them beforehand (before this summer break) to let them know of your situation and most will be understanding.