Given that railway electrification costs are still not under control, I very much doubt we will ever see any 25kV converison of third rail lines at all.
Even in Europe conversion of 1500V or 3kV systems to 25kV has been essentially abandoned on the grounds of huge cost.
(France now investigating replacement of 1500V with 9kV, other places talking of conversion from 1500V to 3kV).
In reality the service densities on Basingtoke-Southampton will never get high enough to prevent third rail being upgraded to provide the power required. You'd be spending billions for negligible benefits.
Advances in cable technology (we now have semi practical superconducting feeder cables undergoing test in the Japanese railway industry and in more general grid use elsewhere) are likely to reduce the disadvantages of third rail over time. A superconducting return cable would cut losses dramatically, improve safety by reducing running rail touch voltages and largely eliminate galvanic corrosion concerns. And such cables are under test.
EDIT: Cables being installed at
Gare Montparnasse