The independence and multivariate normality assumptions immediately imply
$$
\begin{pmatrix}
x_1 \\
\vdots \\
x_m \\
y_1 \\
\vdots \\
y_n
\end{pmatrix}
\sim
\mathcal{N}_{\left(m+n\right)d}
\left(
\begin{pmatrix}
\mathbf{1}_m \otimes \mu_1\\
\mathbf{1}_n \otimes \mu_2
\end{pmatrix}
,
I_{m+n} \otimes \Sigma
\right).
$$
Hence,
$$
\bar{x} - \bar{y} \sim \mathcal{N}_d \left(\mu_1 - \mu_2, \frac{m+n}{mn} \cdot \Sigma \right)
$$
and
$$
\frac{mn}{m+n} \cdot \left(\bar{x} - \bar{y}\right)^\top \Sigma^{-1} \left(\bar{x} - \bar{y}\right) \mathrel{=:} T_a \overset{H_0}{\sim} \chi^2\left(d\right).
$$
If $\Sigma$ is unknown we can start from
$$
m \cdot S_x + n \cdot S_y \sim W_d\left(\Sigma, m + n -2 \right),\\
S_x=\frac{1}{m}\sum_{i=1}^{m}\left(x_i - \bar x\right)\left(x_i -\bar x\right)^\top, \\
S_y=\frac{1}{n}\sum_{i=1}^{n}\left(y_i - \bar y\right)\left(y_i -\bar y\right)^\top,
$$
where $W_d\left(\Sigma, m + n -2\right)$ denotes the Wishart distribution with scale matrix $\Sigma \in \mathbb{R}^{d \times d}$ and $m + n -2$ degrees of freedom.
Since $S = \left(m \cdot S_x + n \cdot S_y\right)/\left(m + n\right)$ is independent of $\left(\bar{x} - \bar{y}\right)$, we get
$$
\frac{mn\left(m + n - 2\right)}{\left(m + n \right)^2} \cdot \left(\bar{x} - \bar{y}\right)^\top S^{-1} \left(\bar{x} - \bar{y}\right) \mathrel{=:} T_b \overset{H_0}{\sim} T^2\left(d, m + n - 2\right),
$$
where $T^2\left(d, m + n - 2\right)$ denotes the Hotelling $T$-squared distribution with $d$ and $m + n - 2$ degrees of freedom. Note that $S$ is the MLE for $\Sigma$ (see, e.g., this derivation which can readily be extended to our setting).
The connection to the $F$-distribution is given by
$$
T_b \sim T^2\left(d, m + n - 2\right) \iff \frac{m + n -d - 1}{d\left(m + n - 2\right)} \cdot T_b \mathrel{=:} T_c \sim F\left(d, m + n -d -1\right),
$$
where $F\left(d, m + n -d -1\right)$ denotes the $F$-distribution with $d$ and $m + n -d -1$ degrees of freedom.
With that you can proceed as usual and compare the realized value of $T_a$ or $T_c$ to the $\left(1-\alpha\right)$-quantile of the $\chi^2\left(d\right)$ or $F\left(d, m + n -d -1\right)$ distribution, respectively, to carry out a hypothesis test at a $\alpha \cdot 100\%$ significance level.
