Gavin Jackson, head of Europe, the Middle East and Africa at Amazon Web Services, loves to talk about snowballs. Not the lumps of mush and ice that children chuck at each other, but Amazon’s portable information storage devices, big grey suitcases that hold huge amounts of data.

When clients such as banks sign on with Amazon Web Services, the ecommerce juggernaut’s cloud-computing service, they upload encrypted data from their old legacy IT systems into the snowball, or the larger-capacity snowball edge. These are then shipped...