How to fix when water is not sinking from bathroom and balcony?

It’s my first question on this site, even though I have toured it but still if my question is not to the conventions of this site please let me know and I shall try to improve it.

I live on the 3rd floor in my building, yesterday when I got home after work and went to wash myself in bathroom (I want to stress that it’s not toilet but the bathroom) I found that water is not sinking into the sink rather it is still there and my bathroom has become a pool. The first step that I did was the obvious one, I poured acid (acid is colloquial term for that hard toilet cleaner) into the sink, pierced some pointed materials into it’s holes (my sink looks like this, it is at the corner of my bathroom), and finally used the bathroom suction cup. But unfortunately nothing worked well, it then stroke me that the sink is clear but the drainage pipe might be clogged. I came out and found that my balcony had water coming out from it’s sink, it became quite a mystery of how it could happen. Let me give you a sketch of room (the bathroom is attached)

And after contacting to other society members, I found that people just below me (that is on flats immediately below me on 2nd and 1st floor) have no such drainage problem. Considering that the water is not going down the drainage pipe but flowing out of the balcony sink seems to suggest that the pipe line from my bathroom sink and pipe line from balcony’s sink get connected to the main drainage pipe and at the joint there is a clog and hence instead of water flowing down it gets transferred to the adjacent pipe (that is from bathroom to balcony and vice versa). After talking to the owner of building and the plumbers, I can present this picture of pipe lines:

(Those red lines are the connecting pipes, connecting the sinks to the main drainage pipe). The joint seems to be clogged and because of that water from bathroom sink flows through the red pipe (red is just for reference to thee picture) and due to blockage doesn’t go to down but flows through the red pipe connecting the balcony’s sink.

I know the last option that is left to me is to call the high-tech plumbers, they will cut the walls and the floors to go to the joint and will find something. But this process is long, it will take 3 to 4 days or even longer with lot of cost and destruction of well arranged house (cutting of wall, and deconstruction of bathroom floor). Is there something that I can do that will not involve the deconstruction and solves my issue? Is it possible to suck out all the water that is stagnant in pipe by some suction pump and then pouring some hard cleaning liquid that will fix the scum?