Battery life calculation using Nano, NRF24L01+ and window magnetic reed switch
You should use interrupts for the reed switch and then put the microcontrolller into sleep for a keep alive interval. Your master device should check that the sensor is still active and set off the alarm if its not. You would also want to allow for a battery low condition that might disable the alarm for convince. This still should keep the power consumption low.

The nano is roughly 1mA @ 3.3v while the CPU is running and 72uA when its in sleep. Discounting power consumption in the alarm condition triggered by the interrupt using an interval of 9 seconds using delay or interrupt timers then average consumption then about a second to wake up and send a keep alive over the RF. (0.000072 * 9 + 0.001 * 1) / 10 = 0.0001648 amps. (average over 10 second duty cycle)

My guesstimate of your RF power consumption without knowing what kind of protocol would be somewhere around 128 bytes per transmission taking about 90ms. (0.0001 * 9910 + 0.015 * 90) / 10,000 = 0.0002341 amps.

Total of their averages is: 0.0003989 or 398.9uA.

Alkaline AA cells vary widely. So instead based on some Ni-MH with 2400mAh per cell and 1.2v

3 cells would be 3.6v so you'll "get" most of the mAhs out of the batteries.

2400mAh = 2400000uAh

2400000uAh / 398.9uA ~= 6016.54 hours or 250 days or 8 months.

Power consumption were intentional overestimates. You should be able to make it last a lot longer.

https://www.avrprogrammers.com/howto/atmega328-power
http://harizanov.com/2013/05/nrf24l01-power-consumption-footprint/