Here is the simplest circuit with 3 parts only. R2 is your relay that closes with delay and R1*C1 determines the delay, so if we simplify and disregard the input current to base, the delay would be proportional to R1.
I used two power sources for simulation but they can be merged into one.
Picture #1 is the schematic, picture #2 is the voltage over time graphs and ZIP contains LT spice schematic so you can play with it.
This circuit is just a hint. Massage it in the SPICE software and determine your values.