Have the switch operate the motor directly and a relay through a series diode, and connect an electrolytic capacitor and low value resistor in series across the relay coil.
When you operate the switch, the relay operated directly through the diode, and the capacitor quickly charges through the resistor.
When the switch is released, the capacitor holds the relay coil in for some time & teh diode stop the power from the cap feeding back to the motor that runs direct from the switch.
The relay contact can be used to switch power to the second motor.
The capacitor value depends on the relay resistance & voltage, you need to try eg. 1000uF and see how that goes.
The resistor should be low to charge the cap quickly, but not so low as it takes excess current from the power source.
It can also be done with transistors, but that's just slightly more complex especially if you need a "sharp" cutoff of power to the second motor.