We quickly decided that the Wemos D1 Mini was the way to go. With WiFi, lots of RAM and the capability to drive WS2812 directly, that's the perfect match for our project.

We then fiddled around with cloud-architecture. To quickly sum it up:

You'll need hot glue. Lots of it.
Styrofoam is okay, but doesn't like hot glue at all. Cardboard works much better, and healthier too.
Take care which batting you use, and preferrably buy local to be able to inspect beforehand.
The best topology for the LED carrier is a torus, if you don't wan't to spend hours soldering LEDs together. It also looks very nice, even without the cloudy batting.

As for the software:

FastLED is a very nice library for doing animations with LEDs.
ArduinoJson is a very nice library for parsing json on the Wemos. It also comes with a great assistant: https://bblanchon.github.io/ArduinoJson/assistant/
OpenWeatherMap has a very nice API, and lets you do 1 query per second on the free tier - enough for hundreds of clouds.