No. So many things wrong!
You do not use regulators on battery projects. You might use switchmode "buck" converters if there is a real need to do so, but certainly not series regulators.
The servos run from 5 or 6 V. That is four alkaline "AA" batteries. Arguably, if you use four NiMH batteries, you can use that for the servos, and also power the Nano.
You do not need to connect the switch to both 5 V and ground. You connect it to ground, and use INPUT_PULLUP mode on that Nano pin. In most cases, the 10k resistor is quite unnecessary.
The big (well, second biggest after that) design problem is that the "Useless Box" needs to be permanently powered, but draw no power until activated. So you cannot leave the Nano powered and you certainly cannot leave the servos powered as they draw resting current.
I haven't built a "Useless Box" myself, so I am not sure how this is usually managed in the Arduino designs. The simple designs are based purely on a gearmotor and some switches so this is no problem.
Actually, you want a Pro Mini rather than a Nano as you do not want resting current drawn by the USB interface either.