$1\,U = ...... \text{kg}$. How is the mass of an atom or nucleus determined?

(N/A) $1$ unified atomic mass unit $(1\,U) = \frac{1}{12}$ of the mass of a carbon-$12$ isotope $\left(^{12}_{6}C\right)$ including the mass of electrons $= 1.66 \times 10^{-27}\,\text{kg}$.
Generally,the mass of a body is measured by a common (physical) balance.
Large masses in the universe (like planets or stars) are determined based on Newton's universal law of gravitation $\left[m = \frac{Fr^2}{GM_e}\right]$.
To measure very small masses (like an atom),a mass spectrometer is used. In this instrument,the radius of the trajectory is proportional to the mass of the charged particle moving in uniform electric and magnetic fields.