This would be very easy with a small Microchip PIC10F processor, which exist in SOT23, and with a small MOS transistor to switch power of the rest of electronic. The processor is directly connected to power and use only 100nA when in sleep mode, and will wake up with a small button for exemple. So, you can do easily precise time delay, as long as you want, using only a few lines of code.
You need only to program the processor, which may need some effort if you are a beginner, anf if you don't have access to a programmer.
It's also possible with analog circuits, but needs some thinking for lowering current and obtain long delay time.
Special circuit like LTC2950 exist, but are not cheap, not easy to find, and don't work with 2 AA cell.