How many neurons is needed for UAT to hold?
There is not a particular number. You have to decide how much you want to approximate your function.

UAT, in informal language, means that neural network can approximate any given function f when given appropriate parameters (which are to be found experimentally; their learnability isn't a question here!).

If you have a function of 3 variables, then a neural network with 20 neurons may approximate it far better than one with the 10 neurons. It is possible that after a certain number of neurons, the difference between the outputs, i.e., (F - f) becomes very less.

I answer this question following Occam's razor, thus not mentioning any un-necessary terms. Please ask, if you have any additional or specific questions in it.