Delay time is very ambiguous - it kind of implies that a filter will delay all signals (at any frequency) the same amount and this isn't true. Consider the following picture: -
The two graphs top and bottom to the left show the (frequency response) bode plots of a 2nd order low pass filter with various damping ratios. They do not directly give an indication (to the untrained eye) of how the filter (or system) might respond to a step input.
The step-input-response is shown on the right. Clearly, for different values of zeta (damping ratio) the "response" is very different. Take the example when \$\zeta\$ = 2. The output gradually rises towards 1 - clearly there is some form of delay going on but how much is it or, put another way, what critera do we impose that gives us a time delay value?
Maybe we say the delay is the time taken to reach the 90% level but, this can't apply to the scenarios when \$\zeta\$ is significantly less than 1 because of overshoot and undershoot.
I go along with @LvW and think you should consider talking about group delay.
However, this isn't a simple matter either. Consider a bessel filter and a butterworth filter and how group delay looks: -
The bessel filter is well-known for having a really flat group delay but its filtering characteristics are not as good as a butterworth filter which, as you can see has quite a poor group delay. Basically, what I'm labouring to say is that any system or filter does not have a fixed delay.