The increasing demand for tabular data analysis calls for transitioning from manual architecture design to Neural Architecture Search (NAS). This transition demands an efficient and responsive anytime NAS approach that is capable of returning current optimal architectures within any given time budget while progressively enhancing architecture quality with increased budget allocation. However, the area of research on Anytime NAS for tabular data remains unexplored. To this end, we introduce ATLAS, the first anytime NAS approach tailored for tabular data. ATLAS introduces a novel two-phase filtering-and-refinement optimization scheme with joint optimization, combining the strengths of both paradigms of training-free and training-based architecture evaluation. Specifically, in the filtering phase, ATLAS employs a new zero-cost proxy specifically designed for tabular data to efficiently estimate the performance of candidate architectures, thereby obtaining a set of promising architectures. Subsequently, in the refinement phase, ATLAS leverages a fixed-budget search algorithm to schedule the training of the promising candidates, so as to accurately identify the optimal architecture. To jointly optimize the two phases for anytime NAS, we also devise a budget-aware coordinator that delivers high NAS performance within constraints. Experimental evaluations demonstrate that our ATLAS can obtain a good-performing architecture within any predefined time budget and return better architectures as and when a new time budget is made available. Overall, it reduces the search time on tabular data by up to 82.75x compared to existing NAS approaches.