For Google Play, we are working on updating the GPGS plugin. It’s going to have a lot more things it will do including multiplayer features. I can’t provide an ETA on it, but it feels like it’s getting close.
For Apple, you have Game Center which should support turn based games now. If you want real-time, there is a GameKit community plugin that will handle the full Game Center experience (except for videos).
Then you have the option of creating your own service. Many people would have recommended Parse.com, but Facebook us shuttering it at the end of the year. However Facebook open-sourced their code, so you could in theory run it it on your own servers. There is a plugin for talking to Parse, but since parse isn’t being maintained, there likely won’t be updates to the plugin.
I don’t want to write a history book here, but Corona Developer @develephant created Coronium, a Parse like service that you can self-host. That way you don’t have to worry about someone pulling the plugin on you. Corona Labs has hired @develephant to build a cloud based service for Corona SDK users, but there is a lot of work needed to make it work. He has just announced the beta of the self-hosted version https://docs.coronium.cloud if you want in early. Again it’s doesn’t have turn-based features built in, but you could add them fairly easily and it’s all in Lua. Since it’s in beta it’s not recommended for going into production right now, but if you even started today, it will take some time for you to build it out and hopefully it will be production ready before you need it to be. Also this is a community invited project and I’m confident that someone will roll out a module for turn based support for it, but again you’re looking at some time before it happens.
Then you have the option of using Photon. https://www.photonengine.com/en-US/Photon
They have a plugin for Corona. It’s real time, but you should be able to do turn based with them too.