I have contacted Wingo. They recommended that I insert a SIM card from Swisscom (e.g. from a friend) into my smartphone, register it and then change the SIM card again. This should work, as the mobile phone provider is only required for registration.

According to our information, this is actually not supported in Switzerland at the moment. The easiest thing to do is to contact Wingo directly. They can confirm whether this watch works with their network.