What is the minimum pumping length of the following languages?
I understand that a pumping length for a language $K$ is a constant $c_K$ such that each string that has at least that language can be pumped within the language. Here "pumped" can remain somewhat abstract: it may be regular or context-free "uvwxy" pumping for instance.

It is clear that we can take $c_{K\cup L}\le \max\{c_K,c_L\}$. Any string in $K\cup L$ of at least this length can be pumped within either $K$ or $L$ depending or where it comes from.    

It seems we cannot say more than that. The main reason is that we can pump one language into the other. Look at a variant of your last example. If $K$ is finite, say it contains all words shorther than $m$, then $c_K = m$, as no word in $K$ can be pumped. Let $L$ be the complement of $K$ containing all words of length at least $m$. Then $c_L = m+1$ (assuming that pumping incluses "pumping down" where we obtain shorter strings). However $c_{K\cup L}=c_{\Sigma^*}=1$.

I am afraid that for concrete examples you have to do a separate analysis.