Optimal quantization for mixed distributions has emerged as a compelling area of study. In this work, we have focused on a mixed distribution formed from two uniform distributions with partially overlapping supports. For this class of distributions, we have examined the structure of optimal sets of $n$-means and the corresponding $n$th quantization errors for all positive integers $n$. Initially, we explicitly determined the optimal sets and quantization errors for $1 \leq n \leq 6$. Subsequently, we established several key lemmas and propositions and proposed an algorithm that facilitates the computation of optimal $n$-means and quantization errors for all $n \geq 5$. Numerical results are also presented to illustrate the application of the algorithm in deriving these quantities. The findings of this study offer valuable insight and serve as a foundation for further research on quantization in the context of mixed distributions with overlapping supports.