Nice back street route running approximately parallel with Beulah Hill. There are a few different options at Streatham Common - this one turns off Covington Way early to avoid the top of the hill and then rejoins the common (paths) lower down. Easiest way to deal with the A23 is on foot - or set off from the A214 turning right, there's plenty of time and space to get clear of the traffic & you're turning back left on to quieter streets before you know it.
Wandsworth council are a bit uptight about cycling in parks (seems to be a Tory thing?), but as long as you're considerate nobody's really going to mind on Tooting common.
Biggin Wood has steps but there is a ramp as well so it's (just about) possible to get a cargo bike through. Three-wheelers will struggle - you can go around the woods down Biggin Hill and back up Bigginwood Road.
"Quietway 5" continues north from Tooting Common to Clapham Common via Cavendish Road, but it's not great really. There's also quite a nice way through to Wandsworth Common from the northern end of Tooting Common - Dornton, Ritherdon, Cloudesdale, Balham Park Road.
One day Wandsworth council will get around to fixing Magdalen Road and the cycle route running parallel with the A214 (West Side, Birdhurst Road etc.), making for nice onward connections to the Wandle (at Earlsfield) and the Thames (at Wandsworth) respectively.