High-impact climate damages are often driven by compounding conditions, such as elevated heat stress arising from combined high humidity and temperatures. To explore future changes in compounding hazards under several climate scenarios, climate emulators can provide light-weight, data-driven complements to Earth System Models (ESMs). Yet, only a few existing emulators jointly emulate multiple climate variables. We introduce MERCURY (Multi-resolution EmulatoR for CompoUnd climate Risk analYsis), a spatio-temporal, multi-resolution emulator designed for compound climate risk analysis. MERCURY employs image-compression-based techniques for memory-efficient emulation and consists of two main modules. The regional module represents the monthly, regional response of a given variable to yearly Global Mean Temperature using a probabilistic additive model, resolving regional cross-correlations. The resulting regional values are then jointly disaggregated to grid-cell level values using a lifting-scheme operator, founded on principles of Discrete Wavelet Transforms. We demonstrate MERCURY on the humid-heat metric, wet bulb globe temperature (WBGT), as derived from temperature and relative humidity emulations. The emulated WBGT spatial correlations correspond well to those of ESMs and the 95 and 97.5 quantiles of WBGT distributions are well captured, with an average of 5 deviation. MERCURY's setup allows for region-specific emulations from which one can efficiently "zoom" into the grid-cell level across multiple variables by means of the reverse lifting-scheme operator. This circumvents the traditional problem of having to emulate complete, global-fields of climate data and resulting storage requirements.