Suppose I have a sequence of functions $f_n: X \to Y$ between metric spaces that converges uniformly to $f$. Then if I change the metric on $Y$ to a Lipschitz equivalent metric, the sequence of functions will still converge uniformly to $f$.
However, I am told that if I change the metric on $Y$ in such a way that does not disturb the induced topology, then $f_n$ may no longer converge uniformly. My question is: is there a simple example of a sequence of functions from $X$ to two homeomorphic metric spaces $Y$, $Y'$ converging uniformly in one but not the other?
EDIT: question modified slightly in response to comments.
$Y$ must not be compact for this to be possible (because compact spaces have unique uniform structure).
You can simply construct an artificial example. For instance, let $X={\bf N}$ and $Y={\bf N}\times [0,1]$, while $Y'$ is $Y$ with metric on $\{n\}\times [0,1]$ scaled by $1/n$ (so that the diameter is $1/n$, you can put the distance between components in any way you want, for example put $2$ whenever two points are in different components). Clearly, $Y'$ is homeomorphic to $Y$ (even by a locally bi-Lipschitz function), but if you take sequence of functions $(f_m)$ with $$f_m(n)=\begin{cases}(n,1) &\textrm{if }m>n\\ (n,0) &\textrm{otherwise}\end{cases}$$ you will have $f_m$ converge uniformly as sequence of functions into $Y'$, but not so into $Y$. You can replace $[0,1]$ here with any nontrivial metric space.
For a different example, take for $Y$ the interval $(0,1]$ with the standard metric, and for $Y'$ the same interval with the metric scaled by a singular function, e.g. for $y_1<y_2$ put $d(y_1,y_2)=y_1^{-1}-y_2^{-1}$, and put $$f_m(n)=\begin{cases}\frac{1}{n} &\textrm{if }m>n\\ \frac{1}{n+1} &\textrm{otherwise}\end{cases}$$ This sequence will be uniformly convergent in the standard metric, but not in the scaled metric.