Adapting Large Multimodal Models (LMMs) to real-world scenarios poses the dual challenges of learning from sequential data streams while handling frequent modality incompleteness, a task known as Continual Missing Modality Learning (CMML). However, existing works on CMML have predominantly relied on prompt tuning, a technique that struggles with this task due to cross-task interference between its learnable prompts in their shared embedding space. A naive application of Low-Rank Adaptation (LoRA) with modality-shared module will also suffer modality interference from competing gradients. To this end, we propose DeLo, the first framework to leverage a novel dual-decomposed low-rank expert architecture for CMML. Specifically, this architecture resolves modality interference through decomposed LoRA expert, dynamically composing LoRA updates matrix with rank-one factors from disentangled modality-specific factor pools. Embedded within a task-partitioned framework that structurally prevents catastrophic forgetting, this expert system is supported by two key mechanisms: a Cross-Modal Guided Routing strategy to handle incomplete data and a Task-Key Memory for efficient, task-agnostic inference. Extensive experiments on established CMML benchmarks demonstrate that our method significantly outperforms state-of-the-art approaches. This highlights the value of a principled, architecturally-aware LoRA design for real-world multimodal challenges.