How can you detect a time two times faster, i.e. if all entities in
the universe, you
included, moved and grew old suddenly two times faster, without any knowledge in
science and without sophisticated measuring apparatus?

If every thing suddenly goes 2x faster, then I do not think you'd be
able to notice it easily. The easiest, and at the moment, only way I
could think of would be to use some sort of doppler shift phenomenon.
Assuming the speed of sound or light remains a constant, then a
passing racecar would have a higher pitch than it did before as it
approached you.