Either I'm dimwitted (well, then) as I don't see a number 3 bus from Piccadilly tube station to the Dulwich Wood Park bus stop. I'm guessing bus routes change? These are the directions I received from our apartment rental owner:
I like to take the underground to Piccadilly tube station and then get the No 3 bus right from outside the tube station to right outside the apartment. I can give you better directions closer to the time but it is very easy. This way I get to see the sights of London on my way in as the bus goes around Parliament House etc and all around the west end so it makes it more interesting. You can also do a train but as I say, the tube and bus is the nicest and easiest I find.
Anyway, do you plan routes based on fewer changes or fastest route? I'm concerned I'll get confused with too many changes, like with a bus, which side of the street or something. I'm willing to sacrifice a bit of time for the sake of simplicity. Then again I'm not exactly the sharpest knife in the drawer (according to dear old dad).