Why do our hands feel warm when we rub them?

The frictional heat generated in the contact area between our hands flows roughly equally to each hand. The interface gets hotter, and there is a heat flow by conduction away from the interface in both directions. At the interface itself, there is a discontinuity in the heat flux, with opposite signs to the two fluxes. The difference between the two heat fluxes is the frictional heat generation rate per unit area. So we have $$\left[-k\frac{dT}{dx}\right]^+-\left[-k\frac{dT}{dx}\right]^-=q_{gen}$$So, $$\left[-k\frac{dT}{dx}\right]^+=\frac{q_{gen}}{2}$$and$$\left[k\frac{dT}{dx}\right]^-=\frac{q_{gen}}{2}$$

As a result of all this, the temperature is highest at the interface (higher than the bulk of your hands), and this is what you sense as your skin temperature.

As long as the nerve endings and the skin cells in its neighbourhood aren’t in equilibrium, there will be a sensation of heat. This of course is an oversimplification because there’s the case of sensitivity.

One possible explanation is that one of the hand gets slightly warmer than the other and hence there is a heat flow from one to other and hence we feel hot.

You almost got it there! Zeroth law is true for bodies in equilibrium. However, when heat is generated by rubbing hands together, the heating in each hand is uneven. This is obvious because the roughness of the hands are uneven and they are in motion. And the unevenness is present between the layers of your skin as well. This creates a gradient in the temperature in your hand. So each hand in itself has regions that are slightly warmer than the others.

But as soon as equilibrium is reached (smoothness in temperature finer than our resolving power), we can no longer feel the heat.

When we rub our hands their temperature rises due to the friction.

Correct. Energy is transferred to the surface of the skin (epidermis) due to friction work which elevates the temperature of the epidermis of both hands.

Now if there is equal increment in the temperature of both the hands then there should be no flow of heat (zeroth law of thermodynamics).

If the temperature of the epidermis of each hand is the same, then theoretically there would be no heat transfer between the hands. But there is heat transfer within each hand from the epidermis to the dermis layer of the skin, discussed below.

Hence we should feel no difference than usual as there is no heat, rather due to there being a temperature gradient heat should flow out of our hands to surrounding and our hands should feel cooler (assuming that surrounding has same temperature as our normal body temperature).

The reason our hands feel warmer is the elevated temperature at the surface of the skin, the epidermis, due to the friction work, is greater than the normal temperature of the dermis located below, so that heat transfers to the dermis where the nerve endings sensing heat are located.

Hope this helps.