OK Steve.
I'm sorry about the delay. I lost track of this project.
This design consists of two monostables (pulse generators) and is built around an NE556, which contains two NE555 timers, probably the most famous IC among hobbyists. The first, on pins 1~6 of the 556, is triggered by the pushbutton (SW1), and generates a pulse on relay K1, which drives part of your display (I don't think you gave details).
When this pulse ends, the second monostable is triggered, and it generates a shorter pulse on relay K2, which drives the motor.
The first monostable is triggered when pin 7 goes below about 1/3 of the power supply voltage, which happens when pushbutton SW1 closes. RB and CB provide filtering to prevent triggering due to interference coupled into the input.
The duration of the first pulse is approximately 1.1 x C1 x (VR1 + R1) where C1 is in farads and VR1 and R1 are in ohms. VR1 provides an adjustable resistance between zero ohms and about 100 kilohms, and R1 acts as a "stopper resistor" so there is always some resistance in that circuit, even when VR1 is at its "MIN" end. (You need a stopper resistor of at least 1k to prevent damage to U1.)
So with VR1 at "MIN", the total resistance will be 22k (R1 only). Since C1 is 82 microfarads, or 0.000082 farads, the first monostable's pulse duration will be 1.1 x 22000 x 0.000082 which is about 2 seconds. With VR1 at "MAX", the total resistance will be about 122k so the pulse duration will be about 11 seconds.
The actual maximum time may be up to 20% longer or shorter than that, because potentiometers usually have a +/- 20% tolerance on their end-to-end resistance. There is also inaccuracy in C1 and I have suggested three alternative components. The middle one is an 82 uF aluminium electrolytic with a +/- 20% tolerance, and the other two are Tantalum capacitors with tighter tolerance but different capacitance values; the time range will change in proportion to the difference in capacitance.
As well as initial value error, temperature and aging will affect the capacitance of these capacitors somewhat. Tantalum capacitors may be more stable than aluminium electrolytics. Using capacitors like this is a compromise for simplicity. It's possible to get much more accurate timing using a crystal-controlled circuit but this needs a lot more components. Let me know if you want to look into that possibility.
The first monostable's output on pin 5 goes high (+12V or somewhat lower) during the first monostable's time; this voltage activates relay K1. The relay type I've suggested has a fairly low coil current (below 20 mA); the 556 can drive up to 200 mA from its outputs.
When pin 5 returns low, the network of RS, CT and RT generates a low-going pulse on pin 8, which triggers the second monostable. The second monostable operates exactly like the first, and activates relay K2. Its timing is controlled by C2, a 15 uF capacitor, so its output pulse is about 5.5 times shorter.
I've shown each of VR1 and VR2 as a potentiometer, i.e. a user-adjustable variable resistor with a shaft for a knob. If you don't need user-accessible adjustment, you could use a preset potentiometer (aka a trimpot) for either or both.
The contacts of the relay I've suggested are rated for 3A at 250VAC and 3A at 28V DC. The motor is an inductive load and should have a diode connected backwards across it, rated for at least as much current as the motor draws. For example, a 1N4001 is rated for 1A. This is needed to protect the relay contacts from "back EMF" which is a high voltage spike that can be generated by the motor when power is removed due to its inductive nature. The two 1N914 diodes across the relay coils are there for the same reason (the relay coils are inductive).
CD is a decoupling capacitor to suppress voltage spikes due to relay switching and switching within U1.
I've shown the power input and relay output connections on a 6-pin connector, and the pushbutton just connected straight up to the circuit. I imagine you'll want to change this to suit your requirements. I'm sure you know how to deal with the mains voltages on K1 safely.
Any questions feel free to ask.
Again my apologies for the delay in getting this ready.