Can the cyclicality of the real wage be used as a measure for the degree of competiveness in the labour market? When reading the answers to this question of mine, I'm left wondering if cyclicality of real wages can be seen as an indicator for the nature and degree of competition in the labour market. 

In perfect competition, we would have high procyclical wages, under monopsony acyclicality.

What would be the cyclicality, if the labour market could resemble a oligopoly(a highly unionized market could resemble an oligopoly, right?), or if the labour market worked as if in monopolistic competition?
In general, no. The procyclicality of wages can also It could also be a measure of labor market frictions instead of the level of competition. 

For an example that highlights the difference, consider an economy where only union electricians can do electrical work and where the union controls the members and their hours. If the union can frequently and costlessly adjust their total labor supply, then they don't have a lot of friction in their supply but they (by construction) have limited competition through their monopoly labor supply.  On the other hand, laws mandated costly separation processes for employees such as 2 years severance or, more extremely, lifetime employment make it very difficult to adjust the quantity of labor supplied but this isn't exactly a competition issue. There could be many employees, many employers, and no licensing restrictions and yet this could still significantly dampen the cyclicality of the real wage by making employment not adjust to cyclical productivity shocks. 

This might be somewhat splitting hairs because both restrict the adjustment of labor demanded but conceptually they are quite different. 

Additionally, there may be reasons related to other features of the economy or the way that the data is gathered for wages to be less procyclical in one place than another. For example, measurement error attenuates statistical relationships towards zero so in two otherwise identical countries, if one has wage measured with more error then the correlation between wage and your favorite business cycle measure will be shrunk towards zero in that country relative to the truth.