The Royal Hospital for Neuro-disability in Wandsworth (assuming it has an open day);
The Horniman Museum and Gardens in Forest Hill.
Having finally escaped various one-way systems, I found myself at Eltham Palace.
The exterior viewed from the gardens:
Although occupied since at least Anglo-Saxon times, the site of the current palace became a royal residence from 1305, when it was acquired by the future Edward II as a moated manor house. Successive monarchs added to the buildings and gardens, and organised a variety of tournaments and other entertainments there. Then, in the 1470s, Edward IV added the Great Hall that has survived into the 21st Century, along with a service range of which three timber-framed gables survive.
Not much is recorded of what Richard III thought about the palace, but Henry VII used it as a nursery for the princes and princesses, and Henry VIII continued to visit in the early years of his reign -- before deciding he preferred Hampton Court. After that, no one seems to have bothered with the place much although Elizabeth I is reported to have visited and James I made disparaging reports about its state of repair. Charles I also visited, but he was the last monarch to do so.
During the Civil War, Parliamentarians stayed in the palace and were even less complimentary about it than James I. After that, the palace became alternately a picturesque ruin and a farm (with the Great Hall used as a barn). All would have been lost in the 1820s, until the Marquis of Lansdowne and allies mounted a preservation campaign in the House of Lords. The Great Hall was saved, and became a tennis court for gentlemen's residences built around the site. These survived until the 1930s, when Stephen and Virginia Courtauld took out a 99 year lease on the site and started an ambitious plan to build a modern residence that incorporated the Great Hall (but not the 19th Century additions).
The architects Seely and Paget, aided by Sir Charles Peers who had been an inspector of ancient monuments prior to WWI and had been involved with preserving the Great Hall in 1910-11, built a 'restrained' exterior inspired by Wren's Hampton Court Palace. The Courtaulds then went a bit over the top with the interiors, installing lots of useful modern conveniences, but also what I consider to be a quite excessive amount of Art Deco elements for a house (not that the interiors would have looked entirely out of place in an office block, a top-end furniture store or a cruise liner, but this is supposed to be a family home). Decor aside, the Courtaulds lived in the palace (with a pet lemur) from 1936 to 1944 (when they moved out due to 'no longer being able to get the staff' to maintain the place). It was then rented out to the Army School of Education until 1992 when English Heritage took over, having already been responsible for the Great Hall since 1984.
Fortunately for my readers' brains, photography is not permitted inside the palace, although I would have liked to show you the interior of the Great Hall and its Edward IV themed stained glass. Also of note (and pictured in the English Heritage Guidebook written by Michael Turner, MVO) are the 1930s coin-operated telephone for the use of guests; the flower room with its cupboards, sink, and shelves of many different vases; the leather map of the palace and grounds in the library; and Virginia Courtauld's bathroom.
Photography of the outside, of course, was perfectly okay, although there were a fair number of other visitors wandering around and cluttering up some of my shots (hence why I usually go out and about on weekdays).
Great Hall viewed from the main gardens:
Great hall and part of 1930s building viewed from gardens around 1930s turning circle:
1930s entrance and Edward IV's timber-framed gables:
Site of 1930s swimming pool:
Remains of older building works:
Two views from the bridge over the moat on my way out:
For more information and far more interior features than I've mentioned, I can highly recommend Michael Turner, MVO's Guidebook, ISBN 9781848020900, available here.