A logical way to calculate the induced magnetic field would be to start with EMF, find the current in the loop, bearing in mind the self-inductance and the resistance of the loop. Once you have the current in the loop, you can use Biot-Savart law to calculate the induced magnetic field.
Your question about the inducing a secondary EMF, etc will, I think, be addressed by the self-inductance. This is why it is difficult to establish current through inductors, like loops. The induced magnetic field resists the imposition. Self-inductance of simple shapes like loops is known.
A short-cut would be not to deal with magnetic fields (since they may not be uniform), but to only work with fluxes. This is useful since even if your applied magnetic field is uniform, the magnetic field generated by induced current will not be. So let the self-inductance of your loop be $L$, resistance $R$. Let the applied magnetic flux be $\Phi_a$ and let the corresponding EMF be $\mathcal{E}=-\dot{\Phi}_a$.
Connecting the applied EMF, as a voltage source, in series with self-inductance and resistance, I would then expect a dynamic system characterized by the following equation, which you solve for current $I$:
$$
RI+L\dot{I}=\mathcal{E}=-\dot{\Phi}_a
$$
The induced flux would be $\Phi_i=LI$, so that in absence of resistance there would be zero net flux through the loop.
I think the solution would be:
$$
\begin{align}
I\left(t\right)=&\frac{1}{L}\int^t_{-\infty}dt'\,\exp\left(-\frac{R}{L}\cdot\left(t-t'\right)\right)\,\mathcal{E}\left(t'\right)\\
=&-\frac{1}{L}\,\int^t_{-\infty}dt'\,\exp\left(-\frac{R}{L}\cdot\left(t-t'\right)\right)\,\dot{\Phi}_a\left(t'\right)
\end{align}
$$
This is of course an approximate approach that only makes sense in the magnetostatic limit, i.e. when the size of the loop divided by the rates of change of current is much smaller than the relevant speed of light
