In multi-class histopathology nuclei analysis tasks, the lack of training data becomes a main bottleneck for the performance of learning-based methods. To tackle this challenge, previous methods have utilized generative models to increase data by generating synthetic samples. However, existing methods often overlook the importance of considering the context of biological tissues (e.g., shape, spatial layout, and tissue type) in the synthetic data. Moreover, while generative models have shown superior performance in synthesizing realistic histopathology images, none of the existing methods are capable of producing image-label pairs at the same time. In this paper, we introduce a novel framework for co-synthesizing histopathology nuclei images and paired semantic labels using a context-conditioned joint diffusion model. We propose conditioning of a diffusion model using nucleus centroid layouts with structure-related text prompts to incorporate spatial and structural context information into the generation targets. Moreover, we enhance the granularity of our synthesized semantic labels by generating instance-wise nuclei labels using distance maps synthesized concurrently in conjunction with the images and semantic labels. We demonstrate the effectiveness of our framework in generating high-quality samples on multi-institutional, multi-organ, and multi-modality datasets. Our synthetic data consistently outperforms existing augmentation methods in the downstream tasks of nuclei segmentation and classification.