I don't think you can do quite what you specified with only one relay.
If you are prepared to modify your requirements so that the circuit is reset when the ignition is turned off (rather than off and back on again) then the solution is simple.
simulate this circuit – Schematic created using CircuitLab
Figure 1. The modified circuit with all switching in positive lines.
Figure 2. By moving the load positive to bypass the ignition switch the OP's circuit timing and operation will match that of Figure 1.