The time delay can be increase by increasing the value of the capacitor (Cx). In the schematic a 100uF electrolytic capacitor is used in the circuit that will give you a time delay of around 2 minutes. I also checked the circuit with a 220uF capacitor and it gave me around 5 minutes delay. You can also experiment with the other values.
The circuit is also using an LED with a 470 Ohms current limiting resistor for indicating the ON and OFF condition of the relay switch. If you want to make the circuit simpler you can also extract the LED and the 470 ohms resistor from the circuit.
The figure below shows a very simple and useful project / schematic of a time delay relay circuit using a 555 timer IC. The circuit is able to trigger the relay from few seconds to few minutes after pressing the push switch S1. The circuit is quite simple and using only few external parts with the IC.