First notice that $u_n$ only belong to some of the $W^{1,p}$ spaces, as we have $u_n'=\cos (nx)+\frac{1}{\sqrt{x}}$ and
$$\cos(nx)+\frac{1}{\sqrt{x}}\in L^p(0,1)\iff \frac{1}{\sqrt{x}}\in L^p(0,1)\iff |x|^{-p/2}\in L^1(0,1)\iff p<2 $$
Therefore it is only meaningful to discuss convergence in $W^{1,p}(0,1)$ for $p\in [1,2)$.
There is no pointwise limit for $u_n'(x)=\cos (nx)+\frac{1}{\sqrt{x}}$ because there is no pointwise limit for $\cos(nx)$ except for $x=0$. The sequence keeps on oscillating around $\frac{1}{\sqrt{x}}$.
Your Banach-Alaoglu + reflexivity argument works to prove the existence of a weakly converging subsequence if $p\in (1,\infty)$. The issue is that it says nothing about the weak limit aside from existence.
To understand the weak limit, there is a general result (see my answer here) - if $g:\mathbb{R}\to \mathbb{R}$ is a bounded periodic function such that its mean over a period is $\alpha$, then if $v_n(x):=g(nx)$ we have $v_n\rightharpoonup \alpha$ weakly-star in $L^{\infty}(\mathbb{R})$ for $p<\infty$. In this case, we have $g(x)=\cos x$ whose mean over a period is $\alpha=\int_0^{2\pi}\cos x\, dx = 0$. Therefore, $\cos (nx)\rightharpoonup 0$ weakly-star in $L^{\infty}(\mathbb{R})$, hence $\cos (nx)\rightharpoonup 0$ weakly-star in $L^{\infty}(0,1)$ and thus (since $L^{\infty}(0,1)\hookrightarrow L^p(0,1)$ for $p\in [1,\infty]$) we also have $\cos (nx)\rightharpoonup 0$ weakly-star (and hence weakly, since they are reflexive) in $L^p(0,1)$ for all $p\in (1,\infty)$. Finally, we can include $p=1$ as weak convergence in $L^p(0,1)$ for $p>1$ implies weak convergence in $L^1(0,1)$.
In conclusion, the above argument shows that the sequence $u_n$ is weakly convergent to $2\sqrt{x}$ in $W^{1,p}(0,1)$ for all $p\in [1,2)$.
To study strong convergence, notice that as we have proved, the weak limit of $\cos (nx)$ is $0$. Therefore, since the strong limit must agree with the weak limit when it exists, by contradiction if $u_n'\to u$ strongly in $L^p(0,1)$, then we would have $\cos (nx)\to 0$ strongly in $L^p(0,1)$ and in particular in $L^1(0,1)$, but if $2(k+1)\pi\geq n\geq 2k\pi$, then
\begin{align*}\int_0^1|\cos (n x)|\,dx&= \frac{1}{n}\int_0^n|\cos (y)|\,dy\geq \frac{1}{2(k+1)\pi}\int_0^{2k\pi}|\cos y|\,dy\geq \\
&\geq \frac{k}{2(k+1)\pi}\int_0^{2\pi}|\cos y|\,dy=\frac{2}{(1+1/k)\pi}\not \to 0
\end{align*}
a contradiction.
Best Answer
$L^2(\Omega)$ is a subspace of $H^{-1}(\Omega)$, the dual of $H^1(\Omega)$. Thus, a sequence converging weakly in $H^1$ converges weakly in $L^2$ to the same limit.
In general, nothing can be said about strong convergence in $L^2$.