That should work, no problem.
However if small size is important it would be better to use an Arduino that runs on 3v and then you could power the whole thing from a pair of AA alkaline cells. The nRF24 needs 3v in any case. I don't think you can get a 3v Nano but you can get a 3v Micro.
Another possibility (and they way I do it) is to use an Atmega 328 running at 8MH using its internal oscillator and mounted on some stripboard (vero board).
This Simple nRF24L01+ Tutorial should help get you started.
Wireless problems can be very difficult to debug so get the wireless part working on its own before you start adding any other features.
Be aware that if two of your Arduinos transmit at the same time both messages will be garbled. One way to deal with that is to have the master poll the slaves in turn using the concept in my second example. It can easily do that several times per second.
...R