Assume that we have a signal to be detected using the binary testing hypothesis. The received signal is distributed as follows.
\begin{align} \mathcal H_0:& \ x = w\\[2ex] \mathcal H_1:& \ x = s + w \end{align} where $w\sim \mathcal N(\mu_w,C_w)$ and $s\sim \mathcal N(\mu_s,C_s)$
I was trying to apply the NP lemma, but I'm getting stuck somewhere.
You're looking at a Generalized Gaussian Detector$^{\ \text [1]}$. The Neyman-Pearson detector decides $\mathcal H_1$ if: $$ \frac{p(\mathbf x; \mathcal H_1)}{p(\mathbf x; \mathcal H_0)}> \gamma\tag{1} $$ With $\mathbf w$ and $\mathbf x$ independent; you get here: $$\mathbf x\sim \begin{cases} \mathcal N\left(\mu_w, \mathbf C_w\right) &\text{under} \mathcal \quad \mathcal H_0\\[2ex] \mathcal N\left(\mu_s + \mu_w, \mathbf C_s + \mathbf C_w\right) &\text{under} \mathcal \quad\mathcal H_1 \end{cases}$$
The NP inequality in equation $(1)$ gives:
$$ \frac{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_s + C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_s - \mu_w\right)^T\left(\mathbf C_s + \mathbf C_w\right)^{-1}\left(\mathbf x-\mu_s - \mu_w\right)\right]}{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_w\right)^T\mathbf C_w^{-1}\left(\mathbf x-\mu_w\right)\right]}>\gamma $$ Taking the logarithm on both sides gives the needed test statistic $T(\mathbf x)$: $$ \underbrace{\ln\left(\frac{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_s + C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_s - \mu_w\right)^T\left(\mathbf C_s + \mathbf C_w\right)^{-1}\left(\mathbf x-\mu_s - \mu_w\right)\right]}{\displaystyle\frac{1}{(2\pi)^{N/2}\mathrm{det}^{1/2}\left(\mathbf C_w\right)}\exp\left[-\frac{1}{2}\left(\mathbf x-\mu_w\right)^T\mathbf C_w^{-1}\left(\mathbf x-\mu_w\right)\right]}\right)}_{T(\mathbf x)}>\ln(\gamma) $$ You can continue from there. Also, have a look at this related question on NP detector.
$^{\ \text [1]}$: Kay, S. Fundamentals of Statistical Signal Processing: Detection Theory.
