Electrical – switch two 12v sources with 5 pin relay

The following circuit should accomplish what you want:

Normally pin 3 of the 555 timer is off, which keeps the P-channel MOSFET Q1 (high side switch) on, which powers circuit B (output on the left).

When circuit A (input on the left) is powered off, the line going into the trigger input (pin 2) of the 555 is monetarily brought low (C1 and R1 act as a differentiator) and triggers the 555 configured as a monostable. The output pin 3 goes high for 2.5 seconds (determined by the combination of R1 and C2), then pin 3 goes back to 0, turning the MOSFET back on.

The MOSFET is rated at 5A. If for some reason you can't drive circuit B from the MOSFET directly, a lower-amperage MOSFET could be used to operate a relay.

^{simulate this circuit – Schematic created using CircuitLab}

This will almost work. The idea is that the both the motor and the relay is powered as long as the relay is closed. So we've made a latching relay. In position 3, everything is powered, and hopefully it will stay that way as the switch changes to position 2.

There's nothing magic about the ignition switch. Position 0 has nothing connected to it, and position 3 just springs back to 2 when you let go. Position 1 may be wired such that it is on in position 2, but since we don't use it, it doesn't matter.

The problem is that the relay will drop if it breaks the connection while switching from Start to Run. If Run stays energized between position 3 and 2, you're golden. Alternatively, the motor may supply power (as a generator) briefly during the switch return. A third possibility is to put a capacitor across the relay coil, though a fairly large one would be required.