Thanks for the reply. I will just order a new wheel and the riveted part on the basket.
Funny thing is that it usually starts at first (all works ok when it's cold), and then after a short brake i'm unable to do a restart again because of the wheel not going into the holes on the baket, just like in the movie. But I don't think that the temperature should have something to do with this issue.
Here is the image of the old parts before renewal.