Does Wikipedia assume a solution to the halting problem in their description of universal one way functions?
You don't need to know whether it is quadratic on all $x$, only on the particular $x$ that is given as input. And it's easy to decide whether a machine runs in a given time bound on a fixed input $x$. Just simulate it for that many steps and check whether it stops before the time bound runs out.

Here, "at most quadratic" is a little sloppy. It should be read as meaning "at most $|x|^2$ steps", so that there is a concrete time bound to check against. But one should keep in mind that there is a tradeoff between writing that is perfectly rigorous, and writing that can easily be understood by non-specialists, and Wikipedia generally aims for the latter — see https://en.wikipedia.org/wiki/Wikipedia:Make_technical_articles_understandable and its specific guidance to "explain formulae in English".