With the rapid development of online social media platforms, the spread of rumours has become a critical societal concern. Current methods for rumour detection can be categorized into image-text pair classification and source-reply graph classification. In this paper, we propose a novel approach that combines multimodal source and propagation graph features for rumour classification. We introduce the Unified Multimodal Graph Transformer Network (UMGTN) which integrates Transformer encoders to fuse these features. Given that not every message in social media is associated with an image and community responses in propagation graphs do not immediately follow source messages, our aim is to build a network architecture that handles missing features such as images or replies. To enhance the model's robustness to data with missing features, we adopt a multitask learning framework that simultaneously learns representations between samples with complete and missing features. We evaluate our proposed method on four real-world datasets, augmenting them by recovering images and replies from Twitter and Weibo. Experimental results demonstrate that our UMGTN with multitask learning achieves state-of-the-art performance, improving F1-score by 1.0% to 4.0%, while maintaining detection robustness to missing features within 2% accuracy and F1-score compared to models trained without the multitask learning framework. We have made our models and datasets publicly available at: https://thcheung.github.io/umgtn/.