I think that replacing the existing network of 3rd rail (bar Merseyrail deep-level, as the tunnels will be nigh on impossible to electrify for any forseeable benefit) should be a priority, as further wiring of the network is due to take place in the near future. If we have a theoretically fully electrified network (with Southern Region on DC), the costs will be astronomical. All OHLE-slightly less astronomical. Besides, what's the point of all this expensive dual-voltage stock (375s, 376s, 450s, 458s, 444s) if we don't allow them to make use of their capabilities?
Speaking from regular SWT travel, I'm very sure that 110mph running could be achieved with the 450s and 444s if the network was wired around the mainline. There are plenty of straight sections, so should be relatively easy to reach 110. And Southeastern's 375/6 fleet has pantographs they never even USE. We should definitely give them somewhere to run under the wires.