Hi,
I live in Brighton and am often found wandering along the South Downs Way with my dog.
In all my walks, with backpack & camping gear (10-14kg) it takes me about 25 mins to walk a mile, including rest for snacks, photos and chatting. I can walk faster if necessary, but I don't like to be in too much of a rush.
I have walked it in stages, camping and staying in B&Bs and travelling from Brighton then back after a walk.
The walk from Winchester train station to the SDW start shouldn't be seen as an extra hike - it's a beautiful 'city' to walk through, and it's downhill if I remember
I got the train from Brighton and walked from Winchester to Meon Springs where there is a camp site right on the path, so a detour off the path at East Meon is not necessary.
I did this a long time ago - it was actually a trial walk with my pack and camping gear, my first with my dog to see how things went.
I carried way too much and I was not that fit, but still managed to get to Meon Springs in good time.
I then walked to South Harting, which is down off the escarpment, but not too far. I stayed in a B&B and ate in a pub.
I then did a half day walk to Cocking and got the bus to Chichester and train back to Brighton.
I found the walk easy going and chatted to lots of folk on the way. The SDW has great wide, flat paths where you can walk and admire the views without having to concentrate on the path.
I would take a few trips to Box Hill, near Dorking to practice walking up and down hills.
Also think about the cost of travelling back to London (and time) - it may not be much different to paying for a camp site or B&B in one of the many beautiful villages on and around the Downs. YHAs at Truleigh Hill & Southease are excellent.
There are lots of places on the way to get snacks in cafes, say in Queen Elizabeth Country Park, fresh home made ice cream nr Cocking and pubs on route. You won't go hungry or thirsty. There are a few water taps as well in the later stages - Botolphs (A283), Truleigh Hill, Saddlescombe Farm (superb cafe here too), Horsedean Farm (A27), Southease YHA, toilet at Cuckmere Haven.
Good luck
Ian