Why is the heat flow in metals slower than the current flow? When we apply a voltage across a metallic conductor, the current starts to flow almost instantaneously. But when a temperature difference is established across the same conductor, the flow of heat is much slower. It takes larger time for the heat to reach from one end to the other than the current. Why is this so?
 A: The current flows almost instantaneously because it is driven by an electric field which appears across the conductor almost instantaneously (near the speed of light). All electrons in the conductor are set into motion by a chain reaction.  Collectively they all move through the conductor at what is called the drift velocity at the same time.
By contrast, heat transfer by conduction requires the transfer of energy by collisions between particles in the material that starts at the high temperature end of the conductor and progresses gradually to the low temperature end of the conductor. In the case of metals, the particles are primarily electrons.
Hope this  helps.
A: Let me first point out that heat is not carried by electrons. In fact, the temperature of the electron gas is much higher than that of the metal itself (thousands of Kelvins), but it makes a small fraction of the total thermal energy.
Electric current is a response to the electric field, which propagates with the speed of light through the metal. It however takes time before this field is screened by the mobile electrons, since it involves physical movement of matter.
Heat transfer occurs via the interactions between the lattice ions, i.e. purely due physical movement. The equivalent of the speed of light here is the velocity of the lattice phonons.
As a useful fact, it is worth mentioning that the drift velocity of the electrons is much smaller than the velocity of their thermal motion. While Drude model and the Newton's equation with viscous friction seem to result in the same formula for the conductance, they describe rather different situations.
Correction and update
My answer above has incorrectly stated that the heat conductance in metals is due to movement of the lattice ions. This is true for semiconductors/insulators, but not for metals, where the heat also is carried by the electrons. Credit to @thermomagneticcondensedboson for pointing it.
The main point however remains essentially the same: heat conductance is a diffusive process, with its diffusion constant determined by the thermal speed of the particles and the characteristic collision time. On the other hand, electric current is a response to the application of an electric field, which, when turned on, propagates through the metal with the speed of light. On the other hand, the distribution of electrons which results in screening of this field and establishing a steady current-carrying state is a diffusive process, similar to the heat conduction, with a similar time scale.
A: In metals the thermal conductivity is determined by the "free" electrons, that is the electrons with energy with a range around $E_F$ of order $kT$. In the diffusive regime the length over which the heat conduction takes place exceeds the mean free path, which is of order of 50-100 nanometers. The heat diffusion constant is determined by this number and the Fermi velocity. The result is that heat diffuses at a "speed" much smaller than light $c/n$.
At ranges of the order of the mean free path the heat transport is ballistic and occurs much faster.
There is also an effect which does propagate at $c/n$ and that is the appearance of a thermoelectric voltage across the metal. See https://en.wikipedia.org/wiki/Seebeck_coefficient
https://www.semanticscholar.org/paper/Electron-mean-free-path-in-elemental-metals-Gall/5fde93380cf4f2b5564c042d34b5610be16d18c3/figure/1
