Washing machine pressure switches directly control the hot and cold water inlet solenoids and so are rated for quite high currents - 15A at 230V AC is typical. So for low-voltage DC applications, the switches can certainly cope with (say) 5A. This means that, for most loads, a relay won't be needed.
The master timer is realised with monostable IC 555 (IC1) and its ‘on’ time is decided by the resistance of 1-mega-ohm potmeter VR. A 47-kilo-ohm resistor is added in series so that even when the VR knob is in zero resistance position, the net series resistance is not zero.