I believe that the result holds quite generally. The specific case of complex bordism is discussed in the following two papers:
- Larry Smith - On the finite generation of $\Omega_\ast^U(X)$ (J. Math. Mech., 1969)
- Pierre Conner & Larry Smith - On the complex bordism of finite complexes (Publications Mathématiques de l'IHÉS, 1969)
The first paper can be found at the webpage for the Indiana University Mathematics Journal (http://www.iumj.indiana.edu/) while the second paper can be found on NUMDAM (http://www.numdam.org/). Note that there is no projective dimension requirement: $\Omega_\ast^U(X)$ is a coherent $\Omega_\ast^U$-module for any finite complex $X$.
But more generally: any ring spectrum $\mathbb{E}$ induces a homological functor
$$ \mathbb{E}_*(-) : \text{SH}^\text{fin} \rightarrow \mathbb{E}_\ast\text{-grMod}$$
from the stable homotopy category of finite spectra to the category of graded $\mathbb{E}_*$-modules ( where $\mathbb E_\ast = \pi_\ast(\mathbb E) = E_\ast(\mathbb S)$ is the coefficient ring of $\mathbb{E}$).
I believe that the collection of those finite spectra $X$ such that $\mathbb{E}_*(X)$ is coherent as a graded $E_\ast$-module can be shown to be a thick triangulated subcategory of the stable homotopy category of finite spectra using properties of coherent modules and the fact that $\mathbb{E}_*(-)$ is a homological functor.
If $\mathbb E_{\ast}$ is a coherent ring then this thick triangulated subcategory contains the sphere spectrum. But since the thick triangulated subcategory generated by the sphere spectrum is the whole of $\text{SH}^\text{fin}$ it follows that $\mathbb E_\ast(X)$ is a coherent $\mathbb E_*$-module for every finite spectrum $X$.
In other words, if $\mathbb{E}$ is a ring spectrum whose coefficient ring is coherent then $\mathbb E_\ast(X)$ is a coherent graded $\mathbb{E}_\*$-module for any finite spectrum $X$.