In one direction, a rapidly branching tree will have very high doubling dimension, while having topological dimension $0$ (or $1$, if you include the edges). In another direction there is a bound, and this is discussed in the nice paper below (on the first page):

<cite authors="Le Donne, Enrico; Rajala, Tapio">_Le Donne, Enrico; Rajala, Tapio_, [**Assouad dimension, Nagata dimension, and uniformly close metric tangents**](http://dx.doi.org/10.1512/iumj.2015.64.5469), Indiana Univ. Math. J. 64, No. 1, 21-54 (2015). [ZBL1321.54059](https://zbmath.org/?q=an:1321.54059).</cite>