We study a time-inhomogeneous SDE in $\R^d$ driven by a cylindrical Lévy process with independent coordinates which may have different scaling properties. Such a structure of the driving noise makes it strongly spatially inhomogeneous and complicates the analysis of the model significantly. We prove that the weak solution to the SDE is uniquely defined, is Markov, and has the strong Feller property. The heat kernel of the process is presented as a combination of an explicit `principal part' and a `residual part', subject to certain $L^\infty(dx)\otimes L^1(dy)$ and $L^\infty(dx)\otimes L^\infty(dy)$-estimates showing that this part is negligible in a short time, in a sense. The main tool of the construction is the analytic parametrix method, specially adapted to Lévy-type generators with strong spatial inhomogeneities.