Does the existence of instantons imply non-trivial cohomology of spacetime?
First, the reasoning in the question about isomorphism classes of bundles is wrong, because the $\check{H}^1(M,G)$ from the linked math.SE post is not the cohomology of $M$ with coefficients in $G$, but actually the Čech cohomology of $M$ for the sheaf $\mathscr{G} : U\mapsto C^\infty(U,G)$.
However, this indeed has a relation to the cohomology of $M$ itself for $G = \mathrm{U}(1)$, via $$ 0 \to \mathbb{Z} \to \mathbb{R} \to \mathrm{U}(1) \to 0$$ which turns into $$ 0 \to C^\infty(U,\mathbb{Z})\to C^\infty(U,\mathbb{R}) \to C^\infty(U,\mathrm{U}(1))\to 0$$ since $C^\infty(M,-)$ is left exact and one may convince oneself that this particular sequence is still exact since the map $C^\infty(M,\mathbb{R})\to C^\infty(M,\mathrm{U}(1))$ works by just dividing $\mathbb{Z}$ out of $\mathbb{R}$. Considering this as a sheaf sequence $0\to \mathscr{Z}\to \mathscr{R} \to \mathscr{G} \to 0$, $\mathscr{Z} = \underline{\mathbb{Z}}$ for $\underline{\mathbb{Z}}$ the locally constant sheaf since $\mathbb{Z}$ is discrete, and the sheaf of smooth real-valued functions on a manifold is acyclic due to existence of partitions of unity, so taking the sheaf cohomology one gets $$ \dots \to 0 \to H^1(M,\mathscr{G}) \to H^2(M,\underline{\mathbb{Z}})\to 0 \to \dots$$ and thus $H^1(M,\mathscr{G}) = H^2(M,\underline{\mathbb{Z}}) = H^2(M,\mathbb{Z})$ where the last object is just the usual integral cohomology of $M$. Hence, $\mathrm{U}(1)$ bundles are indeed classified fully by their first Chern class which is physically the (magnetic!) flux through closed 2-cycles, and the existence of non-trivial $\mathrm{U}(1)$-bundles would imply non-trivial second cohomology of spacetime (or rather of one-point compactified spacetime $S^4$ since one should be able to talk about the field configuration "at infinty" and the bundle being framed at infinity). Indeed, since $H^2(S^4) = 0$, the existence of $\mathrm{U}(1)$-instantons would contradict the idea that spacetime is $\mathbb{R}^4$.
For general compact, connected $G$, it turns out the possible instantons are pretty much independent of the topology of $M$ because a generic instanton is localized around a point, as the BPST instanton construction shows - the instanton has a center, and one may indeed imagine the Chern-Simons form to be a "current" that flows out of that point, giving rise to a nontrivial $\int F\wedge F$.
Topologically, one may understand this by imagining $S^4$, and giving a bundle by giving the gauge fields on the two hemisphere, gluing by specifying a gauge transformation on the overlap of the two, which can be shrunk to $S^3$, i.e. the bundle is given by a map $S^3\to G$, and the homotopy classes of such maps are the third homotopy group $\pi_3(G)$, which is $\mathbb{Z}$ for semi-simple compact $G$. Since the "equator" can be freely moved around the $S^4$, or even shrunk arbitrarily close to a point, this construction does not in fact depend of the global properties of $S^4$, it can be done "around a point".
Thus, instantons in general do not tell us anything about the topology of spacetime.
This answer has been guided by the PhysicsOverflow answer to the same question.