I'm learning this topic right now, so I'm not expert, but actually, we can say a lot about $\widetilde{SL}(2,\mathbb{R})$. I'll write about three points of view, one group theoretic, one differential, and one Riemannan.
- Group theory
First of all there is a short exact sequence of groups [see Scott, p. 464, for a reference about this discussion]
$$0\to\mathbb{Z}\overset{i}{\to}\widetilde{SL}(2,\mathbb{R})\overset{p}{\to} PSL(2,\mathbb{R})\to0$$
and since $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm1\}$ is just a two-fold cover, i think we also have a s.e.s.
$$0\to\mathbb{Z}\overset{i}{\to}\widetilde{SL}(2,\mathbb{R})\overset{p'}{\to} SL(2,\mathbb{R})\to0$$
which correspond to the multiplication $\mathbb{Z}\overset{\cdot 2}{\to}\mathbb{Z}$.
Anyways, let $G$ be either $SL(2,\mathbb{R})$ or $PSL(2,\mathbb{R})$, and by a little abuse of notation $p:\widetilde{SL}(2,\mathbb{R})\to G$ any of the two projections.
These are central extensions (even though I don't know how to prove it), so there exist a function (of sets!) $$\phi:G\times G\to \mathbb{Z}$$
such that $\widetilde{SL}(2,\mathbb{R})$ is isomorphic to the set $\mathbb{Z}\times G$ endowed with the product law
$$(a,f)\cdot_\phi(b,g):=(a+b+\phi(f,g),fg).$$
Of course there are too many possible functions $g\times G\to \mathbb{Z}$, and only some of them will give rise to $\widetilde{SL}(2,\mathbb{R})$, but all those that do it are characterised by the fact that, if we choose a section (of sets!) $s:G\to\widetilde{SL}(2,\mathbb{R})$ such that $p\circ s=id_G$, then
$$i(\phi(f,g))=s(f)s(g)s(fg)^{-1}.$$
These last part is just general theory about abelian extensions [see for example Brown, IV.3, or this thread], but I don't know how write explicitly such a function $\phi$.
- Differentiable manifolds
Another interesting approach, which I think is related even though I don't know how [see these slides as a sloppy reference], is the following.
The KAN (or Iwasawa) decomposition of $SL(2,\mathbb{R})$ gives us the following isomorphism (of smooth manifolds, but not of groups since, for example, the only normal subgroup of $SL(2,\mathbb{R})$ is $\{\pm1\}$)
$SO(2)\times AN(2,\mathbb{R})\cong SL(2,\mathbb{R})$
where $S^1\cong SO(2)=\Big\{\begin{pmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\\\end{pmatrix}:\theta\in\mathbb{R}\Big\}$ and $\mathbb{E}^2\cong AN(2,\mathbb{R})=\Big\{\begin{pmatrix}r & s \\ 0 & r^{-1}\\\end{pmatrix}:r,s\in\mathbb{R}\Big\}.$
It follow that, as an analitic manifold, it's universal cover is
$$\widetilde{SL}(2,\mathbb{R})\cong \mathbb{R}\times AN(2,\mathbb{R})$$
- Riemannian geometry
I finish pointing out that one might be tempted to write $\widetilde{SL}(2,\mathbb{R})\cong \mathbb{R}\times \mathbb{E}^2$ which for example tells us that $SL(2,\mathbb{R})$ is aspherical, since its universal cover is contractible, but it is in fact even more interesting to write $\widetilde{SL}(2,\mathbb{R})\cong \mathbb{R}\times \mathbb{H}^2$, where $\widetilde{SL}(2,\mathbb{R})$ get the structure of a line bundle over $\mathbb{H}$ of constant curvature $1$. [for this Riemaniann point of view see Thurston, 3.8].
In fact $\widetilde{SL}(2,\mathbb{R})$ is one of the eight $3$-dimensional geometries.
A final note:
I don't know if, and in case how, all this blabbering can be extended to the universal cover of $SL(n,\mathbb{R})$, and I'd be glad if anyone could fill the gaps and questions that I opened up.