Is it possible to upload a Lua script to a NodeMCU using the Wifi interface instead of serial?
The tutorials and examples that I have found all use the serial interface, i.e. a cable, to program the NodeMCU, but I would like to change the program without connecting anything (using a smartphone or a browser)
Yes it is possible. It is kind of a home-brewed option but it works to a certain extent. The only limitation is in size, of course but other than that it works pretty well. Take a look at:
http://www.instructables.com/id/ESP8266-WiFi-File-Management/
You need to have a way to write the PHP program (I wrote it in C#) if you can't write the code in another language you can download and reuse what this user wrote and use your own PHP server and you should be good to go.
If you have questions, please ask.