In high-dimensional time series analysis, Gaussian approximation (GA) schemes under various distance measures or on various collections of subsets of the Euclidean space play a fundamental role in a wide range of statistical inference problems. To date, most GA results for high-dimensional time series are established on hyper-rectangles and their equivalence. In this paper, by considering the 2-Wasserstein distance and the collection of all convex sets, we establish a general GA theory for a broad class of high-dimensional non-stationary (HDNS) time series, extending the scope of problems that can be addressed in HDNS time series analysis. For HDNS time series of sufficiently weak dependence and light tail, the GA rates established in this paper are either nearly optimal with respect to the dimensionality and time series length, or they are nearly identical to the corresponding best-known GA rates established for independent data. A multiplier bootstrap procedure is utilized and theoretically justified to implement our GA theory. We demonstrate by two previously undiscussed time series applications the use of the GA theory and the bootstrap procedure as unified tools for a wide range of statistical inference problems in HDNS time series analysis.