Does it have something to do do with the fact that the individual isn't concentrating on reacting to the stimulus, so it takes longer for it to be processed? How can this be out in terms of neurones and neurological pathways? what is happening for there to be a delayed reaction time?
When one processor is running two processes, A and B, it is not running them simultaneously: it is switching between them. The switching between A and B takes time---hence the delay.
I am not sure which sort of answer is expected. The minimal answer to that would be that the more distracted you are the longer it needs to sample the stimulus space of interest as you are sampling another distracting stimulus simultaneously with your limited sensory capabilities.
Cognitive tasks get just more complex with distraction as the brain cannot know which features (of the full stimulus space from all the sensory modalities) to take into account and therefore needs to sample more data to find out which of the inputs matter for the specific task. In most of natural situations the inputs match together in terms of experienced stimulus combinations (visual field and sound while walking for example) except for surprising situations that require to alter ones attention (a sound that does not match to the visual scenery is unexpected and might be important to check for danger). To decide whether the system can ignore such parts of the input requires computation and computation takes time. I don't think it is useful to tackle the question in terms of delayed signaling along specific neurons or pathways. The signaling itself is more or less of constant speed and not directly altered by distraction. Of course on a very basic level an increased adrenaline level or similar due to distraction might affect the "weights" of certain basic pathways as one is pushed into another state. In such cases therefore it requires the system to compensate those changes with respect to the actual task of interest as it might be biased to different reaction scheme that is not useful for the specific task.
Actually our brains are very good in focusing on specific parts of the inputs that are important for a task (i.e. cocktail party problem) but still it needs more time to correct the erroneous parts in the presence of distraction. In terms of a simple Hopfield network that does pattern completion distraction would be like an increase of the noise level which leads to a higher number of iterations required for convergence while the speed of signal transmission from neuron to neuron was not altered.