Calculate time on power on delay
I think this is more what you are looking for: 







simulate this circuit &ndash; Schematic created using CircuitLab

R2 provides a path to discharge the gate charge of M1. R2 is effectively in parallel with R3 etc. so the time will be a bit less as a result. 

R1 limits the charging current of the capacitors so it does not damage the switches or pull the power supply down. 

D2 is a catch diode to deal with the inductance of the relay coil (as M1 turns off slowly anyway in normal operation, so it may not be an issue, but if the gate of M1 was shorted to ground the voltage at the drain would rise high enough to avalanche the MOSFET most likely, which isn't great).