$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Lie{Lie}\DeclareMathOperator\Der{Der}\newcommand\el{\mathrm{ell}}$Let $\mathcal{M}_{g,1^{\infty}}$ denote the space parameterizing the following data; a smooth projective genus $g$ curve $C$, a point $p\in C$ and a formal parameter $z_{p}$ at $p$. If $G:=\Aut(\mathbb{D},0)$ is the group of pointed automorphisms of the (edit formal) disc, then $\mathcal{M}_{g,1^{\infty}}$ is a $G$-torsor over the universal curve $\mathcal{M}_{g,1}$. Note that $G$ is affine and $\mathcal{O}(G)\cong\mathbb{C}[\lambda_{0},\lambda_{1},\lambda_{2},...][\lambda_{1}^{-1}]$, so that a point corresponds to the change of coordinates $z\mapsto \lambda_{0}z+\lambda_{1}z^{2}...$.
Let $\mathcal{W}$ denote the Witt algebra $\Der(\mathbb{C}(\!(z)\!))$, ie the quotient of the Virasoro by its centre. A construction going back to several people provides a transitive infinitesimal action of $\mathcal{W}$ on $\mathcal{M}_{g,1^{\infty}}$, which usually goes by the name Virasoro Uniformization. Moreover one demands that this be compatible with the $G$ action, where the compatibility is with respect to the embedding of $\Lie(G)$ into $\mathcal{W}$.
I can just about follow the construction, which essentially arises from an action of the group ind-scheme $\Aut(\mathbb{D}^{*})$ which acts on tuples above by regluing $\mathbb{D}$ to $C\setminus P$ along the disc, with a twist by the element of $\Aut(\mathbb{D}^{*})$, and then differentiating the resulting map. At the infinitesimal level the key point is that both $\mathbb{D}$ and $C\setminus P$ are rigid. The problem is that I get a bit stuck attempting to convert everything into down-to-earth language.
Question. If we let $\mathcal{C}_{\el}^{\infty}$ be the space parameterizing elliptic curves equipped with a distinguished point and a formal parameter at that point, how do I describe explicitly the resulting vector fields $L_{n}$ arising from the Virasoro construction?
More precisely, we have the $G$-torsor $\mathcal{C}_{\el}^{\infty}\rightarrow\mathcal{C}_{\el}$ over the universal elliptic curve. We can pull this back to a $G$-a on $\mathbb{C}\times\mathfrak{h}$ via the standard analytic uniformization of $\mathcal{C}_{\el}$ to obtain a $G$-torsor on $\mathbb{C}\times\mathfrak{h}$, which is presumably trivial. What are the resulting $\SL_{2}(\mathbb{Z})\ltimes\mathbb{Z}^{2}$-equivariant vector fields generating a copy of $\mathcal{W}$?