The problem I would forsee with that section would be that you have a lot of smaller stations on double track, which would presumably make it very difficult to operate services faster than the current higher speeds on the classic Southern network along side the stoppers. Of the quadruple track sections approaching London from the South, I would have thought that the only section long enough with enough capacity for high speed running would be the Woking - Basingstoke section.
Basingstoke-Eastleigh ought to be four-track anyway, with the possible exception of Winchester station. The intermediate stations, Micheldever and Shawford, might pose problems, but there is a solution. The section through Shawford already has goods lines, so it is just a question of bringing them up to passenger running standards. Between Winchester and Worting Junction, put the fast lines on the outside. That also helps with the new tunnels, since they could both be single bore, they could be taken at 125 without the need to worry about the shockwave problem. However, it would mean crossing the down fast on the flat to access Micheldever Oil Terminal. Beyond Eastleigh, four tracks from St Denys through Southampton Airport Parkway to St Mary's Junction would be pushing it. It's possible, but difficult. Increasing speeds from there on would be difficult, not just because of the stations, but there are some difficult curves. Acceleration would be the key here, and 25kV should help with that.