Delay circuit with capacitor and transistor doesn't work as expected
It's hard to tell from your schematic what the voltages might be. What is the relay pullin voltage?
I imagine your simulation does not take into account the pickup voltage of the relay. For example, most 5V relays will actually pickup at about 3V or less, a 12V relay might pickup at 4-5V.

One potential way to ensure a higher pickup voltage on your C3 charge curve is to use a series Zener to offset the pickup voltage (though it's a kludge).

There are major problems with the schematic you have:


The output stage has very limited base current drive, and any current drawn by the BJT base will tend to sop the cap charging. 
Even at low current requirements the half wave rectifier results in a large ripple voltage on C1. This may cause relay chatter as it picks up. 
The second RC stage provides very little time extension since it's charging along with the first stage. 


I would not attempt to provide a timer in this fashion, but if you are trapped into this type of circuit for some reason, here is one that might work:







simulate this circuit &ndash; Schematic created using CircuitLab


Changing to a full wave bridge reduces the ripple.
Increasing C1 reduces the ripple.
M1/M2 act as a long tailed pair, and R3-R4 set the transfer point.
D1-R6 provide a more rapid discharge for C2 when the power is off.


The waveforms are like this:



Notice there is still considerable ripple IMO, but much less than the half wave circuit. 

A more reliable way to implement timing requirements like this is with a comparator (or even a 555 timer) to provide positive switching of the relay. Given the packages available, it would end up no more complicated than the above circuit.