In general, if one knows the universal cover of a space, how does one find the universal cover of a homotopy equivalent space? Is there a systematic way to do this? If not, are there certain techniques in practice? Does it become easier if 1 space is a deformation retract of the other?
We have from Exercise 8 in Section 1.3 in Hatcher that homotopy equivalent spaces have homotopy equivalent universal covers. But I don't know if this helps in practice to find a universal cover.
(I'd really appreciate if you could give a few examples of finding the universal cover by finding the universal cover of a deformation retract.)
Suppose $X$ and $Y$ are homotopy equivalent and $\rho\colon U \to Y$ is a universal cover. What Tyrone is hinting at is that if $f\colon X \to Y$ is a homotopy equivalence then $f^*U$ is a universal cover of $X$.
Here is a simple example:
We know the universal cover of $S^1$ is $\mathbb{R}$, and we can take the covering map $exp\colon \mathbb{R}\to S^1$ given by $exp(t) = e^{2\pi i t}$. Now consider the open annulus $A = \{ z\in \mathbb{C}\mid \frac{1}{2} < |z| < \frac{3}{2} \}$, which is homotopy equivalent to $S^1$ via the inclusion $\iota \colon S^1 \hookrightarrow A$ and the normalization map $\eta\colon A \to S^1$ given by $\eta(z) = \frac{z}{|z|}$.
Consider the pullback
$$ \eta^*(\mathbb{R}) = \{ (z, t) \in A \times \mathbb{R}\mid \eta(z) = exp(t)\}. $$
As I mentioned above this is a universal cover of $A$, and in this particular case you can show that the map $\Phi\colon \eta^*(\mathbb{R}) \to (\frac{1}{2}, \frac{3}{2})\times\mathbb{R}$ given by $\Phi(z, t) = (|z|, t)$ is a homeomorphism.