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EDIT: As Vladimir Chernousov has explained me, theorems 1 and 2 are valid over any perfect field $k$, not necessarily a number field. It suffices to prove that if $N$ is a connected reductive $k$-group, then any cohomology class $\xi\in H^1(k,N)$ can be split by a field extension of bounded degree. We may assume that $N$ is a split group. Then by Steinberg's theorem (Theorem 11.1 in "Regular elements of semisimple groups", republished in Serre's "Galois Cohomology"), $\xi$ comes from some torus $T\subset N$, and hence, can be killed by an extension of bounded degree.

EDIT: As Vladimir Chernousov has explained me, theorems 1 and 2 are valid over any perfect field $k$, not necessarily a number field. It suffices to prove that if $N$ is a connected reductive $k$-group, then any cohomology class $\xi\in H^1(k,N)$ can be killed by a field extension of bounded degree. We may assume that $N$ is a split group. Then by Steinberg's theorem (Theorem 11.1 in "Regular elements of semisimple groups", republished in Serre's "Galois Cohomology"), $\xi$ comes from some torus $T\subset N$, and hence, can be killed by an extension of bounded degree.

Proof of Theorem 1. Choose an element $g\in G({\bar{k}})$ such that $g\cdot H_1\cdot g^{-1} =H_2$. Let $s\in\mathrm{Gal}({\bar{k}}/k)$, then $^s\!g\cdot H_1\cdot \,^s\!g^{-1}=H_2$. Set $c_s=g^{-1}\cdot \,^s\!g$, then $c_s\in N({\bar{k}})$, where $N$ is the normalizer of $H_1$ in $G$, and $c=(c_s)\in Z^1(k,N)$. Let $[c]\in H^1(k,N)$ denote the cohomology class of $c$.

Let $b\in \mathrm{Br}(k)_n$. We may assume that $k$ has no real embeddings. We know from the class field theory that $b$ defines local invariants $\mathrm{inv}_v(b)\in\frac{1}{n}\mathbb{Z}/\mathbb{Z}$ for all finite places $v$ of $k$, and these local invariants are nonzero only for a finite set $\Xi$ of places. By weak approximation there exists a Galois extension $K/k$ of degree $n$ such that $K_v:=K\otimes_k k_v$ is a field for $v\in\Xi$, where $k_v$ is the completion of $k$ at $v$. Then the extension $K_v/k_v$ kills $\mathrm{inv}_v(b)$ for all $v\in\Xi$, hence the extension $K/k$ of degree $n$ kills $b$. This completes the proofs of Proposition 3, Theorem 2, and Theorem 1.

EDIT: This resut is valid over any perfect field, see the end of the answer.

Proof of Theorem 1. Choose an element $g\in G({\bar{k}})$ such that $g\cdot H_1\cdot g^{-1} =H_2$. Let $s\in\mathrm{Gal}({\bar{k}}/k)$, then $^s\!g\cdot H_1\cdot \,^s\!g^{-1}=H_2$. Set $c_s=g^{-1}\cdot \,^s\!g$, then $c_s\in N({\bar{k}})$, where $N$ is the normalizer of $H_1$ in $G$, and $c=(c_s)\in Z^1(k,N)$. Let $[c]\in H^1(k,N)$ denote the cohomology class of $c$.

Let $b\in \mathrm{Br}(k)_n$. We may assume that $k$ has no real embeddings. We know from the class field theory that $b$ defines local invariants $\mathrm{inv}_v(b)\in\frac{1}{n}\mathbb{Z}/\mathbb{Z}$ for all finite places $v$ of $k$, and these local invariants are nonzero only for a finite set $\Xi$ of places. By weak approximation there exists a Galois extension $K/k$ of degree $n$ such that $K_v:=K\otimes_k k_v$ is a field for $v\in\Xi$, where $k_v$ is the completion of $k$ at $v$. Then the extension $K_v/k_v$ kills $\mathrm{inv}_v(b)$ for all $v\in\Xi$, hence the extension $K/k$ of degree $n$ kills $b$. This completes the proofs of Proposition 3, Theorem 2, and Theorem 1.

EDIT: As Vladimir Chernousov has explained me, theorems 1 and 2 are valid over any perfect field $k$, not necessarily a number field. It suffices to prove that if $N$ is a connected reductive $k$-group, then any cohomology class $\xi\in H^1(k,N)$ can be split by a field extension of bounded degree. We may assume that $N$ is a split group. Then by Steinberg's theorem (Theorem 11.1 in "Regular elements of semisimple groups", republished in Serre's "Galois Cohomology"), $\xi$ comes from some torus $T\subset N$, and hence, can be killed by an extension of bounded degree.

Since there are only finitely many conjugacy classes of connected semisimple subgroups in $G_{\bar{k}}$, see Friedrich Knopp's answer to this question, we see that Theorem 1 follows from the next Theorem 2.

We see that we may assume that $N$ is connected (and reductive). Let $R=Z(N)^0$ denote the identity component of the center of $N$, then $R$ is a torus. For a torus the theorem is clearly true (over any perfect field), so it suffices to prove the theorem for the connected semisimple group $N/R$.

Since there are only finitely many conjugacy classes of connected semisimple subgroups in $G_{\bar{k}}$, see Friedrich Knop's answer to this question, we see that Theorem 1 follows from the next Theorem 2.

We see that we may assume that $N$ is connected (and reductive). Let $R=Z(N)^0$ denote the identity component of the center of $N$, then $R$ is a torus. A torus (over any perfect field) splits over a field extension of bounded degree, see YCor's comment, so we may assume that $R$ is split, and then $H^1(k,R)=1$ by Hilbert's Theorem 90. Therefore, it suffices to prove the theorem for the connected semisimple group $N/R$.