Group delay, the derivative of the phase with respect to frequency, is mathematically equivalent to time delay. At RF we use a network analyzer to plot group delay. Don't know what test instrument you would use at audio frequencies.
The easyest way is to simulate a step function, i.e. a squarewave with a VERY long period, but measure the distance between edges at the input and output of the amp.
In practise it's not that stringent - a relatively low frequency, say a few tens of Hz up to a few kHz is fine. The step function method requires a storage or sampling scope, the simplified one can be used with a regular scope, though a digital one is preferable. Trigger input is always on the input waveform, and you look at the output and input alternatively. You still require a high bandwidth scope with a low jitter trigger.
This test is really a sort of cheat. It avoids phase shifts being undistinguishable from an actual delay by using an input signal with a fundamental well inside the range where the amp's phase shift is negligible. This way the various phase shift phenomena all get lumped into the shape of the output signal edge (rise/fall time, over/undershoot, ringing). The other cheat part is that you have to make an educated guess as to which point of the edge you are using as a reference on the output. Zero crossing may be the most obvious but it is not necesairly so. The problem here is that 1/Ft is likely to be far less than transit time because of dominant pole compensation. In other words, the measurement should really be done open loop, but then the reference point problem becomes very prominent (think unclean edges, asymetric slew rates, differences wrt. output amplitude, etc).