The hardware in Tiny BLE is capable of being a BLE keyboard and mouse, but unfortunately they don't provide source code for this right out of the box. There is a demo project for Tiny BLE using the mbed online compiler (https://github.com/Seeed-Studio/Tiny_BLE), and I also found sample code of a BLE keyboard and of a BLE mouse using mbed (https://github.com/jpbrucker/BLE_HID), but they aren't designed specifically for Tiny BLE. So they need to be merged together.

Unfortunately I didn't find good sample code that allows a single Bluetooth BLE device to behave as both a keyboard and mouse at the same time. The only code I did find that is capable of both at the same time was some code on a forum at https://community.cypress.com/thread/11130 that uses theCypress PSoC Creator graphical tool, and therefore the generated code is quite large and tricky to understand.

So my device will act as either a keyboard or a mouse, but not both at the same time.

Anyway, after a bunch of trial & error (is it just me or is Bluetooth extremely unreliable?!), I got it to work as a keyboard, and then with a code rebuild, as a mouse! I used the free mbed OS and online compiler, because BLE apparently doesn't easily fit into the restricted flash size that free versions of compilers such as Keil and IAR support.