"Street name" is jargon for when a brokerage holds your shares.
Nothing will happen. I started with Waterhouse in 1996. Then it became TD Waterhouse, then TD Ameritrade. Now it will be TD Ameriscottrade, or something like that.
Stocks are like a piece of property. You pay some guy (brokerage) to buy it and store it for you. The guy may die or quit, so a new guy comes in and continues to hold it. The property does not change when that happens.
Now if the underlying company gets bought out, splits, or changes in some other way, then that is a different story.