Just a thought: Schools pretty much in all of Europe (the UK is the exception) have deadlines much later than the US, perhaps in April/May or even later. And what's more, there are no tuition fees (again, the UK is the exception). There are some very good schools for mathematics in Denmark (DTU, Copenhagen University), Sweden (KTH, Chalmers/University of Gothenburg, Lund, Uppsala), France (Paris VI, Bordeaux, etc.) and Germany. I'm just naming a few countries and schools where I would hypothetically be interested in staying.
I guess you would have to know some French or German for the schools in France or Germany respectively, but I know that the Nordic schools are very welcoming to international students and offer courses in English.
If you're really set on going to the US you could try to find a one or two year master's programme in Europe as a springboard. Or you could just stay there for your PhD. Keep in mind that you will often have to have a better idea of what you would like to study when you apply at European institutions compared to American ones. For example, you often apply for a specific PhD position with a specific advisor as opposed to a general "PhD Programme".
I have no idea if visas will be an issue but I can't imagine that they will be if the schools are willing to have you as a student.