, all of the Nano's pins start outputting 3.3V instead of 5V.
What caused you to measure the voltage? I have been using nRF24s for a few years now and it has never occurred to me to measure the voltage.
Have a look at this Simple nRF24L01+ Tutorial.
Wireless problems can be very difficult to debug so get the wireless part working on its own before you start adding any other features.
The examples are as simple as I could make them and they have worked for other Forum members. If you get stuck it will be easier to help with code that I am familiar with. Start by getting the first example to work
There is also a connection test program to check that the Arduino can talk to the nRF24 it is connected to. If the first example does not work be sure to try the connection test for both of your Arduinos. Nothing will work if the connection test fails.
A common problem with nRF24 modules is insufficient 3.3v current from the Arduino 3.3v pin. This seems to be a particular problem with the nano. The high-power nRF24s (with the external antenna) will definitely need an external power supply. At least for testing try powering the nRF24 with a pair of AA alkaline cells (3v) with the battery GND connected to the Arduino GND.