Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure

For each induced bundle chart $\tilde{x}:{\pi }^{-1}(U)\to x(U)\times {\mathbb{R}}^n$, declare the sets ${\tilde{x}}^{-1}(O)$, with $O$ open, to be basic open sets. The transition homeomorphisms from [L1] make these families agree on overlaps, so they define a topology on $TM$ for which every $\tilde{x}$ is a homeomorphism onto the open set $x(U)\times {\mathbb{R}}^n\subseteq {\mathbb{R}}^{2n}$.

In the topology of step 1.1, $\pi$ is continuous because its expression in every bundle chart is projection onto the first $n$ coordinates. Two vectors over distinct base points have disjoint neighbourhoods pulled back from the Hausdorff base $M$; two vectors over the same point have disjoint neighbourhoods in one Euclidean bundle chart. Thus $TM$ is Hausdorff.

By [A1], [F3], and [L2], the cover of $M$ by base-chart domains has a countable subcover. Each corresponding open subset of ${\mathbb{R}}^{2n}$ has a countable basis, and the inverse images of those bases form a countable basis for $TM$. Hence $TM$ is second countable.

By [L1], the induced chart transitions are smooth with smooth inverses. Together with steps 1.1-2.2, the induced charts therefore define a smooth $2n$-manifold structure. The construction uses the maximal smooth atlas of $M$, so the resulting structure is canonical.