For an elliptical orbit in the $z=0$ plane, the equations for the time-dependent position are
where the constants $a$, $b$, and $e$ are the semi-major axis, semi-minor axis, and eccentricity. The $x$-direction here is along the major axis of the ellipse.
The angular quantity $E(t)$ is called the "eccentric anomaly" and is related to the time by “Kepler’s equation”,
where the constant $n$ is called the "mean motion". It’s basically an average angular velocity and is just $\Omega$ for a circular orbit.
Given a time $t$, you have to solve the transcendental equation (3) numerically, or using a series expansion, to get $E(t)$. You then put this into (1) and (2) to get the position at time $t$.
These equations are discussed in Wikipedia.
For an inclined orbit, you can simply apply a three-dimensional rotation matrix.