Timer Relay Logic Question

I am dealing with a relay logic circuit that requires a 5min time delay only if a switch cuts power to that circuit. Ie, upon startup it does not require a 5min time delay but after startup if a switch is tripped (power lost), it requires a 5 min time delay before allowing power to that circuit. A simple timer relay would work for the 5min on-delay but how can I bypass it once upon startup?