There may be a cloud. There's nothing to stop Hammerhead uploading data from the app to SRAM. That's probably what Shimano are worried about.

Absolutely no cloud. All this stuff happens on-bike; it has to, because people frequently ride in places where there is no data service.

(Also, it's not clear how a cloud would help here. What we're talking about is allowing the groupset and the computer to share data in real time, to show on the head unit what gears are selected & etc. A cloud would just impose delay.)

There is no cloud. This is how a bike computer communicates with the shifters on your bike.