It is clear that if a sequence of continuous functions converges pointwise to a discontinuous function, then the convergence is not uniform (since uniform limits of continuous functions are continuous). It is also easy to construct examples of sequences of continuous functions that converge pointwise but not uniformly to a continuous function if the domain of the functions is not compact. (If the domain is unbounded, something like $\frac{1}{n}\cdot x$ works, if the domain is bounded but not closed, an adaption of $x^n$ on $[0,1)$ works.)
On compact domains, it is not so obvious how one could construct such an example, although, as we shall see, it is not difficult to construct examples on compact intervals. Nevertheless, Dini's theorem shows that one must exercise some care.
The classical example of a sequence of continuous functions that converges pointwise but not uniformly to a continuous function consists of functions that are $0$ everywhere except for a triangular spike of constant (or even increasing) height that becomes narrower and moves to one endpoint of the interval. Often, one endpoint of the triangle's base is kept fixed. Let's make a slightly more general construction.
Consider a continuous function $g \colon \mathbb{R} \to \mathbb{R}$ with $g(0) = 0$ and $\lim\limits_{\lvert x\rvert \to \infty} g(x) = 0$. Examples of such functions are $g(x) = \frac{x}{1+x^2}$ or
$$g(x) = \begin{cases}\quad x &, 0 \leqslant x \leqslant 1 \\ 2-x &, 1 \leqslant x \leqslant 2 \\ \quad 0 &, x < 0 \lor x > 2.\end{cases}$$
The latter leads to the triangular spike. Then, for $n \in \mathbb{N}$ define $g_n(x) = g(n\cdot x)$. The conditions on $g$ ensure that $(g_n)$ is a sequence of continuous functions that converges pointwise to $0$, hence to a continuous function, and the convergence is uniform on $\mathbb{R} \setminus (-\varepsilon,+\varepsilon)$ for every $\varepsilon > 0$. But we have $g_n(\mathbb{R}) = g(\mathbb{R})$ for every $n > 0$, so the convergence is not uniform on $\mathbb{R}$, nor on any neighbourhood of $0$, unless $g \equiv 0$.
Here, we want a sequence of continuous functions that converges uniformly on $[-r,r]$ for every $r \in (0,1)$ and pointwise but not uniformly on $[-1,1]$. To achieve that, we move the critical point of the construction from $0$ to $-1$ by a translation, and define $f_n(x) = g_n(x+1)$ for $x\in [-1,1]$. For both explicitly given functions $g$, this construction satisfies the requirements.
In your question, you mentioned that you looked at $\sum x^n$. Power series cannot give examples with the desired properties, for if a power series converges at both endpoints of an interval, it converges uniformly on the whole compact interval.
That series diverges when $x=1$ (since $n>1\implies\frac1{\log(n+2)}\geqslant\frac1n$) and converges when $x=-1$ (by the alternating series test).
Since it converges when $x=-1$, it follows from Abel's theorem that it converges uniformly on $[-1,0]$. Actually, it converges uniformly in $[-1,r]$ for any $r\in(0,1)$, but it does not converge uniformly in $[-1,1)$.
Best Answer
About the original task formulation: The series $$\sum_{n=1}^\infty\frac{nx}{n^2+x^2}=x\sum_{n=1}^\infty\frac{n}{n^2+x^2}$$ does not converge at all, obviating the question about uniform convergence, as for $n>|x|$ you get $$\frac{n}{n^2+x^2}>\frac1{2n}.$$ Thus the harmonic series $$\sum_{n=1}^\infty \frac1{2n}$$ is a diverging minorant, proving divergence.