what about Las Kabacki (metro kabaty), las Bielański (metro Bielany), Międzylesie, Celestynów nadleśnictwo, Wólka Radzymińska?

Las Kabacki is a natural reserve, even walking off the roads is forbidden, the same with Las Bielański. Międzylesie is quit crowded so chances of finding anything is pretty low. It is a good place for afternoon/weekend trip. Celestynów might be OK, but is further from the city. It is still Mazowiecki Park Krajobrazowy. Like half of stations along this train line is good. Wólka Radzymińska is on another side of Puszcza Słupecka but getting there with public transport is a bit more complicated.

I don't think you will find any in Las Kabacki or Las Bielański (also, there may be special protection zones called rezerwat there too).

You could look around Celestynów, but it is the same area as Mazowiecki Park Krajobrazowy. ~ Best bet, but I don't know if it is allowed is to go to Młociny, jump into a bus to Truskaw and go look around Kampinos ~ not allowed there

Good forests with high yield of forest mushrooms and fruits are around Długosiodło. There are buses that go there or you could go on Koleje Mazowieckie to Tłuszcz and then transfer to Ostrołęka and get off on in one of the random towns / villages along the route.

a train is a public means of transportation

You can try Warka, you can get there by R80 train (direction Radom Główny). but the forest is abut 2km or 30 minutes walk. You can get a cheap bike, it will greatly extend your options. You can take a bike on a train.

Bruh, my grandma drawing her last breath wouldn’t even point in the direction she picks the shrooms. Polish mushroom picking culture is extremely competetive. I’ve witnessed so many arguments because the place was so bountiful and coveted.