Dynamic graph learning (DGL) is essential for modelling evolving graph data, but existing methods suffer from significant computational overhead due to repeated full-snapshot retraining and are not well-suited for collaborative settings with partitioned data. In realistic graph systems, cross-partition edges are unavoidable, but direct sharing of graph structure between clients may violate privacy constraints. We propose DG-CoLearn, a client-oblivious collaborative dynamic graph learning framework built on incremental graph snapshot processing, which focuses computation on graph regions affected by temporal updates while preserving historical information through temporal modelling. This incremental design is consistently applied across the entire graph processing pipeline, including a server-mediated embedding exchange mechanism to enable accurate multi-hop message passing without exposing raw cross-client structural information. Extensive experiments demonstrate that DG-CoLearn achieves up to 33.8$\times$ speedup in training time and 27.4$\times$ reduction in communication overhead, while consistently improving predictive performance on both node classification (up to 13.36% F1 improvement) and link prediction (up to 8.27% MAP improvement) tasks. These results highlight the effectiveness of DG-CoLearn in bridging efficiency, scalability, and client-to-client structural privacy in collaborative dynamic graph learning.