In the case of a cartesian product of two (nice enough) topological spaces $X\times Y$ it is known that the universal cover is the cartesian product of the universal covers of the factors. In the more general case of fiber bundles, is there a general method to know what the universal cover is?
Ore specifically, I'm interested in bundles over the circle $S^1$ with fiber a $2$-dimensional compact manifold.
The universal cover of $S^1$ is the real line $R$, if you pullback the bundle $p:P\rightarrow S^1$ over $R$, you have a trivial bundle since $R$ is contractible, so this bundle is the product $R\times S$ where $S$ is the surface. I assume that $S$ is closed, so its universal cover is either $S^2$ or $R^2$, so the universal cover of $P$ is $R\times S^2$ or $R^3$.