Is there cognizance or meta-awareness of classical conditioning? Behaviorism's initial popularity was largely due to it's ability to take cognition and consciousness out of the equation; bringing it close to something that could be tested completely objectively by judging behaviors. Two of the most significant remaining findings of behaviorism are classical and operant conditioning. Because these acts can be studied in terms only of stimulus and reaction, cognition is largely ignored in related studies, especially in the traditional radical behaviorists' work. 

My question is, to what extent is cognition actually at work during classical conditioning? Has it been shown that participants in studies are actually aware that they are starting to associate the conditioned stimulus with either the unconditioned stimulus or the conditioned response?

I'm excluding operant conditioning as I assume that at least for humans it is much more likely that a conscious awareness of the associations between stimuli and punishment/reward will result; perhaps this assumption is incorrect, I just thought it would help narrow the scope of the question. Feel free to correct me if it's wrong, I thought it should probably be a separate question however.

I'll also exclude Conditioned Taste Aversion as it's a rather special case and often only takes a single exposure to stimulus.

Is there evidence of conscious awareness of classical conditioning, especially while the association process is still going on?
I'm not sure what you're asking.  If it's found that there's awareness of the relationships in the experiment then typically it's argued that's not conditioning and the behaviour is modified through insight.  It's been a bone of contention with respect to classical conditioning and humans in the past.  You might want to look at Lovibond and Shanks (2002).  I think some of the thinking there is wrongheaded (how you assess awareness).  Nevertheless, it's a start for looking into the topic.

If you're asking if there's an awareness of the effects of being conditioned separate from those of stimulus and response correlations, I don't think anyone has ever tested that.

Lovibond, P. F. and Shanks, D. R. (2002). The role of awareness in Pavlovian conditioning: Empirical evidence and theoretical implications. Journal of Experimental Psychology: Human Perception and Performance, 28(1):3–26.