I received the WIFI shield from Makerfabs:
In both cases, I was able to connect to it with two Apps on my phone:
On the App, I was able to auto detect the engine and run a few functions. Unfortunately, I wasn't able to connect wirelessly from JMRI. It seemed that the IP address might have changed or something, but the connection was not successful.
I therefore ordered a display to add to the Arduino to display its IP address...