The proof that any Cumulative Distribution Function is Càdlàg within the axioms of Kolmogorov is based on countable additivity.
However, countable additivity is (and this needs to be taken with a pinch of salt$^1$) not present in Cox approach.
Hence my conclusion: that CDF needs not to be Càdlàg within the Cox Probability Theory approach (I might be wrong though).
Of course, the CDF is Càdlàg for a discrete Random Variable, in the Cox approach. But it might not be for continuous RV.
I don't want to upset, but this is, I think, a point that shows that Cox approach can cover cases (namely when the CDF of a continuous RV is not Càdlàg) that Kolmogorov approach can't cover (again, I might be wrong).
$^1$: To be taken with a pinch of salt, because according to gwr in his comment to this post there might be a way to prove countable additivity within the Cox approach.
Book that presents Cox approach to Probability Theory: Jaynes' Probability Theory: the Logic of Science.