In https://en.wikipedia.org/wiki/Uniform_convergence it's given that
A sequence ${f_n}$ of functions converges uniformly to a limiting function $f$ if the speed of convergence of $f_n(x)$ to $f(x)$ does not depend on $x$.
Now the definition for uniform convergence is
Suppose $S$ is a set and $f_n : S → R$ is a real-valued function for every natural number $n$. We say that the sequence $(f_n)_{n \in > \mathbb{N}}$ is uniformly convergent with limit $f : S → R$ if for every $ε > 0$, there exists a natural number $N$ such that for all $x ∈ S$ and all $n ≥ N$ we have $| f n ( x ) − f ( x ) | < ε$.
Now where does it require for the speed of convergence to not depend on $x$?
Does for example $f_n=\frac{1}{\sqrt{n}}$ converge uniformly even if it certainly does not have the same speed of convergence for all $x$?
"Speed of convergence" is a somewhat fuzzy informal term for how fast the necessary $N$ increases as $\varepsilon$ drops towards $0$.
With pointwise convergence each $x$ can have its own dependence between $\varepsilon$ and $N$. On the other hand, uniform convergence requires that you can give an $N$ based on only what $\varepsilon$ is, and that $N$ then has to work for every $x$.
In other words there is a function from $\varepsilon$ to $N$, and this function (intuitively encoding the "speed of convergence") does not depend on $x$.
Uniform convergence does allow the sequence to converge faster than this common function for some particular $x$, so it would perhaps be more precise to say that the "speed of convergence" is bounded rather than to say that it is independent of $x$.