Results from simulations of hierarchical structure formation cosmology provide a tool to develop an evolutionary model of regular magnetic fields coupled to galaxy formation and evolution. We use the dynamo theory to derive the timescales of amplification and ordering of magnetic fields in disk and puffy galaxies. Galaxies similar to the Milky Way formed their disks at $z\approx10$ and regular fields of $\mu$G strength and a few kpc coherence length were generated within 2~Gyr (at $z\approx3$), but field ordering up to the coherence scale of the galaxy size took another 6~Gyr (at $z\approx0.5$). Giant galaxies formed their disk already at $z\approx10$, allowing more efficient dynamo generation of strong regular fields (with kpc coherence length) already at $z\approx4$. Dwarf galaxies should have hosted fully coherent fields at $z\approx1$. This evolutionary scenario and number of predictions of the model can be tested by measurements of polarized synchrotron emission and Faraday rotation with the planned Square Kilometre Array.