The cost-distance Steiner tree problem seeks a Steiner tree that minimizes the total congestion cost plus the weighted sum of source-sink delays. This problem arises as a subroutine in timing-constrained global routing with a linear delay model, used before buffer insertion. Here, the congestion cost and the delay of an edge are essentially uncorrelated, unlike in most other algorithms for timing-driven Steiner trees. We present a fast algorithm for the cost-distance Steiner tree problem. Its running time is $\mathcal{O}(t(n \log n + m))$, where $t$, $n$, and $m$ are the numbers of terminals, vertices, and edges in the global routing graph. We also prove that our algorithm guarantees an approximation factor of $\mathcal{O}(\log t)$. This matches the best-known approximation factor for this problem, but with a much faster running time. To account for increased capacitance and delays after buffering caused by bifurcations, we incorporate a delay penalty for each bifurcation without compromising the running time or approximation factor. In our experimental results, we show that our algorithm outperforms previous methods that first compute a Steiner topology, e.g. based on shallow-light Steiner trees or the Prim-Dijkstra algorithm, and then embed this into the global routing graph.