Uniform limit of uniformly continuous functions

No, in general, the locally uniform limit of uniformly continuous functions is not uniformly continuous. For an example, consider

$$h_n(x) = \begin{cases} -n &, x \leqslant -n \\ x &, -n < x < n\\ n &, x \geqslant n\end{cases}$$

and

$$f_n(x) = h_n(x)\cdot x.$$

Then all $f_n$ are uniformly continuous, and the convergence is uniform on all compact subsets of $\mathbb{R}$, but the limit function $f(x) = x^2$ is not uniformly continuous.

You get a uniformly continuous limit if the sequence is uniformly equicontinuous, for example. (The uniform convergence of $f_n$ on all of $\mathbb{R}$ implies the uniform equicontinuity of $(f_n)$, so that is a special case of this sufficient criterion.)