Do I need a broadband connection in my home to have a smart meter?

No. Smart meters use an entirely separate, bespoke wireless system. You don't need Wi-Fi in your home for it to work and it won't use your Wi-Fi if you have it. Your smart meter and in-home display communicate via a secure national network which is solely for smart meters. This works in the same way as other wireless systems like car remote keys or TVs, using radio waves.