Vauxhall is on the Victoria tube line, which will take you into the heart of London, to Victoria Station. From there you can easily walk to Buckingham Palace, Westminster Abbey, Parliament, Big Ben or switch to another tube line or catch a bus.
St. George Wharf Pier is near Vauxhall, and you can catch the river boat RB6 there. Takes you directly to Westminster Pier.
If this apartment is near the river, facing the river, you will have a fine view.
Nine Elms has a lot of new apartments, cafes and shops being built there. The new U.S. Embassy in London is being built there. Not far away is Battersea Park and Power Station.
Pro: These will be modern apartment buildings with brand new kitchens and baths.
Cons: The buildings (and neighborhood) will lack the character found in London's older neighborhoods. There will be construction still going on in the area, so could be noisy at times; mostly in the daytime when you're out sightseeing anyway.
New Covent Garden Market is in Nine Elms. It's a huge wholesale fruit, vegetable, and flower market.
You would be near Lambeth, and could walk north and then east to The Imperial War Museum. There's a walking trail on the riverside, called The Thames Path. You could use that for your walking and stay off the street until you got closer in to central London. You could walk this all the way up to the London Eye.
If you don't mind, would you please send me a link to this apartment by PM? Just want to see the views and what the exterior of some of the new apartments look like out there. Thanks.