Nope! Just upload it once, and let us know in the order notes that you want us to use that handwriting for the other products, too.
For example, say you want a Gold Bar Handwriting Necklace, a Silver Disc Handwriting Necklace, and a Handwriting Cuff Bracelet, all with the same signature on them.
When you add the first item to your order, upload the signature that one time. When you add the other two items to the order, just type in the instructions that you want to use that same signature.