You can enter the southern part of Nuuksio national park from Siikajärvi. Behind Siikaranta conference hotel is a parking place if you arrive by car. From Siikaranta to the nearest fire site of lake Holma-Saarijärvi is about 3 kilometers by main paths. From Holma-Saarijärvi you can continue to lake Mustalampi via Korpinkierros circle trail and come back to Siikajärvi.
At Holma-Saarijärvi there are two fire sites, one on an island and one at the lean-to. At weekends you will have company at fire sites. There was plenty of wood at the time I visited. One not so nice souvenir of Nuuksio in September is a deer fly, hirvikärpänen in Finnish which is one nasty insect that likes to stick to your hair and most persistent ones you will find at home.
In September weekends there are quite many people, quite many picking mushrooms. If you exit main trails to less used paths which are plenty, there will be no rush. Since mushroom pickers use no trails they will blend in in the forest.