I think all of this is possible, but not with the standard code provided by Pimoroni. Which is ok, their code is always a good starting point but for you own projects you need to be creative.
So I see a number of tasks:
- log data locally to a file on the internal flash
- execute code when you press a button or
- poll for the hotspot (i.e. router)
- configure the network of the Pico to use the phone as a router
- read the file on the internal flash and upload all data
- delete the file on the internal flash again
For all of these tasks you will find code-snippets in the Internet. Putting everything together is work, but you should also learn a lot along your journey.