Fourier transform is an isomorphism...but we don’t get when each frequency appears?
It's true that taking the Fourier transform will leave you without any (visible) information on time and vice versa, but of course you don't lose any information, you just represent it in a way such that in one domain you only see time information, and in the other you only see frequency information.

Take as an example the Fourier transform of a time-inverted (real-valued) function $x(-t)$: its Fourier transform has the same magnitude as the Fourier transform of the original function $x(t)$. The difference between these two Fourier transforms lies exclusively in the phase. So you're right to assume that timing information is encoded in the phase of the Fourier transform.

There is no time localization in the Fourier transform because its basis functions are complex exponentials extending from $-\infty$ to $\infty$. There are other transforms that will give you a certain degree of time and frequency localization, the most well-known of which is probably the Short-time Fourier transform. Also take a look at this related question and its answers.