I know that after a star undergoes the process of mass loss, depending on the mass of the core the stellar remnants gets converted into a white dwarf star, neutron star or a black hole. Hence, if the mass of the stellar remnant exceeds the Tolman–Oppenheimer–Volkoff limit, then it gets converted to a black hole. So is there currently any equation which helps us to calculate the amount of time taken by the leftover core to get converted to a black hole or the rate at which the mass is gravitationally collapsing?
It depends entirely on the circumstances of black hole formation.
The remnant may be a stable neutron star, but if it accreted enough matter later on (by fallback in a supernova, or from a binary companion), then it may collapse. This could happen at any time after formation.
If the core of the collapsing star forms a proto-neutron star above the TOV limit of stability, then no force can prevent black hole formation. It will happen on the cooling timescale of the proto-neutron star at the centre. Once it is "cold" then the collapse would occur on a free fall timescale - which is about a millisecond in these circumstances. The cooling timescale of a proto neutron is governed by how long the copiously generated neutrinos can be trapped in the core, which could be as long as(!) a few seconds