Time shifted unit impulse function in frequency domain
Apply the Time Shifting property of Fourier Transform.

If
$$ x(t) \implies  X(j\omega) $$
then
$$ x(t-a) \implies e^{-j\omega a} X(j\omega)$$
For impulse,
$$ \delta (t) \implies1$$
Means, it contains all frequencies and their magnitudes are unity.

For the time shifted version,
$$\delta(t-a) \implies e^{-j\omega a}$$

This is a complex number with magnitude = 1. Hence the magnitudes of all  the frequency components are still unity in the frequency domain. But their phase angles are different, $$ \phi = -\omega a $$
In the earlier case, the phase \$ \phi \$ was zero for all the frequency components.