Yes, this is true. First prove it for $|x|=1$, then use the homogeneity, and that the degree of homogeneity is $\geq 1$, while $\mathrm{dist}(x,Z)$ is also homogeneous but of degree $=1$.

EDIT. My previous answer was incorrect. So I replace it. The answer is no. A counterexample is $$y^{2m}+(z^{m-1}y-x^m)^2.$$ This is of degree $2m$ but $\delta$ is like $\epsilon^{2m^2}$ near the point $(0,0,1)$.

I found this example in the paper of Kollar and Shiffman, TAMS 329 (1992), on the very first page. They credit it to Lojasiewicz himself, IHES, Bures-sur-Yvette, 1965. I do not know how to find this IHES preprint, but Kollar and Shiffman is available to everyone online.