The one time I messed up and switched to the Google Nexus (Android OS) I just swapped the sim cards and the phone worked out of the box, but there was sync issues with voicemail and mms messages. There were a few days where I went back and forth between the iPhone and the Nexus, and the vm notification and service reliability was sporadic.
Turns out that I had to inform my carrier every time I was making the switch between each device as the data language between both devices had to be toggled on the carrier side so that the device would receive the information from the network to be displayed correctly on the device I was using.
Long story short, swapping cards on the fly between non-iPhones caused the voicemail/sync issues sounds like you're experiencing. Not sure if that's been corrected by now, but this was in Canada around 2014. So not very long ago, but not so recent. Might be something worth exploring.