Willow tree is lovely but I wouldn't recommend it as a site if your main purpose is exploring London. It's not near public transport at all, and the nearest stations are quite far out on their respective lines. It's a 15 minute walk from the bus stop (without pavement) - which is then a 15 bus to the tube - which is the metropolitan line and takes a little while to get to central. https://citymapper.com/london/?star...=51.588594,-0.449625&eaddr=Willow Tree Centre
Frith Grange camp site is a 5 minute walk from Mill Hill East station, which is on the Northern line. It's much nearer than chigwell / willow tree.
Mill Hill East is on it's own weird little shuttle part of the northern line ,so on weekends you need to change at finchley central.
There's also South london Scout camp which is in the south east of london. It's near Sydenham station, so it's trains rather than tube but they do accept oyster on the trains. The train only takes 15 minutes to get to Victoria and is quite frequent and runs till late. http://www.slsc-thefort.org.uk/
It's a lovely camp site. We stayed there for a week with our SSAGO unit. They even washed / dried my laundry for me because I was going to another camp straight after that one!
Chigwell is probably 1 hour into London (bus+central line tube) but is very easy to get to and close to the bus stop.
South London scout centre is about 10-15 mins walk through the woods (longer via the road) to a train station and then about 20 mins on the train to victoria. Its also very close to the girlguiding centenary maze at crystal palace.
If you fancy something different for a weekend, look at the Angel Community Narrowboat based near st Pancras. They provide a skipper and everyone gets to steer the boat. You go on the canal but you can also stop and get off to look around. You can go up to Camden (massive exciting market) and up through little Venice (past London zoo).
London tips - 2for1 offers with train tickets, climb up the monument for £3ish (fun, great views and tonnes cheaper than the London eye), walk along the Southbank - lots going on and its free (and not overly busy), go in the singing lift in the Southbank centre, have a look at the free museums for exhibitions (V&A has fashion based things), lots of free stuff in Greenwich, free views of London from the rooftop of the One New Change shopping centre near st pauls