If heat is supplied to a system consisting of water and ice at 0 degrees Celsius, which will absorb heat first and what is the underlying physics?

If heat is supplied to this system, it is impossible to tell which component (water or ice) will initially absorb the heat. That depends on how the heat was delivered to the system.

However as the system equilibrates, the heat will spread throughout the entire system. Any heat delivered to the ice will first go towards its latent heat of fusion. When the system equilibrates, if there is any ice remaining, then the entire system must still be at $0^{circ},textrm{C}$ by the zeroth law of thermodynamics.

It may be possible for both the water and ice to absorb the heat simultaneously if the system is heated reversibly. Consider the following thought experiment:

Let the ice and water mixture be in a perfectly insulated metal container for a long time so that we can be assured the ice and water are in thermal equilibrium with one another at 0$^o$C.

We now remove the thermal insulation from the metal container and immerse the container in a large thermal bath (constant temperature thermal reservoir) whose temperature is infinitesimally greater than 0$^o$C. This allows us to heat the ice water mixture reversibly. (If necessary, to insure uniform heating, we could include a small stirrer to circulate the water and ice, and include the stirrer work as part of the energy transfer to the mixture).

So since the thermal bath is in thermal equilibrium with the water and in thermal equilibrium with the ice, per the Zeroth law the water and ice should be in thermal equilibrium with one another.

Finally, per the first law and assuming all the ice doesn't melt, since there is no temperature increase the heat transfer from the thermal bath will equal the mass of ice in the mixture that melts times the latent heat of fusion.

Hope this helps.