Question: This is not really a question because I think I have a solution, so I am asking for a review, some opinion for improving and going further, for solving the ambiguous points, as well as a reference.
Let $K$ be an intermediate field of a Galois extension $L/\Bbb{Q}$ with Galois group $G=Gal(L/\Bbb{Q})$, let $L_v$ be the completion of $L$ at a $p$-adic valuation, which comes with an embedding $\iota:L\to L_v$.
$\Bbb{Q}_p $ is meant to be the closure of $\Bbb{Q}$ in $L_v$. With $\mathfrak{q} = \{ \alpha\in O_L,v(\alpha)> 0\}$ then $Gal(L_v/\Bbb{Q}_p)=D(\mathfrak{q})=\{ \rho\in G, \rho(\mathfrak{q})=\mathfrak{q}\}$.
Any $p$-adic valuation on $K$ comes from an embedding $\iota\circ \sigma: K\to L_v$ for some $\sigma\in G$. The corresponding completion is $\overline{\iota\circ\sigma(K)}$ where $\overline{{}^{{}^\quad}}$ means the closure in $L_v$.
The fundamental theorem of Galois theory tells us
$$\overline{\iota\circ\sigma(K)}=\overline{\iota\circ\sigma'(K)}\qquad \iff\qquad Gal(L_v/\overline{\iota\circ\sigma(K)})=Gal(L_v/\overline{\iota\circ\sigma'(K)})$$ Next, the key argument is that $Gal(L_v/\overline{\iota\circ\sigma(K)})$ is the subgroup of $Gal(L_v/\Bbb{Q}_p)$ fixing $\iota\circ\sigma(K)$, that is $$Gal(L_v/\overline{\iota\circ\sigma(K)})=Gal(L_v/\Bbb{Q}_p) \cap Gal(L/\sigma(K))=D(\mathfrak{q}) \cap \sigma Gal(L/K)\sigma^{-1}$$
The $p$-adic completions of $K$ correspond to the set of subgroups $\{ D(q) \cap Gal(L/\sigma(K)), \sigma\in G\}$ and the two completions $\overline{\iota\circ\sigma(K)},\overline{\iota\circ\sigma'(K)}$ are isomorphic iff $$\exists g\in D(\mathfrak{q}),\qquad Gal(L_v/\overline{\iota\circ\sigma'(K)})=g \, Gal(L_v/\overline{\iota\circ\sigma(K)})g^{-1}\tag{1}$$ $$\iff \exists g\in D(\mathfrak{q}),\qquad D(\mathfrak{q}) \cap \sigma' Gal(L/K)\sigma'^{-1}=D(\mathfrak{q}) \cap g\sigma Gal(L/K)\sigma^{-1}g^{-1} $$
Letting $H=\{ h\in G, h(K)=K\} = \{ h\in G, h Gal(L/K) h^{-1}= Gal(L/K)\}$, at first I thought it would reduce to $\sigma'\in D(\mathfrak{q}) \sigma H$ but apparently it isn't true, as this condition means $ \exists g\in D(\mathfrak{q}),\sigma' Gal(L/K)\sigma'^{-1}=g\sigma Gal(L/K)\sigma^{-1}g^{-1}$ which is stronger than $(1)$.
All the $p$-adic completions of $L$ are isomorphic, so the choice of $\mathfrak{q}$ shouldn't affect the result, indeed replacing it by $g(\mathfrak{q})$ amounts to permuting the isomorphism classes of completions.
That said it is unclear to me if the isomorphism class of the completion $\overline{\iota\circ\sigma'(K)}$ is determined solely by $D(\mathfrak{q}) \cap \sigma' Gal(L/K)\sigma'^{-1}$ or if we need to specify $\mathfrak{q}$ as well.