Problem with shower and basin waste

I have replaced a downstairs shower and basin.
The shower and basin were plumbed in to existing plastic connections which are set in concrete and go under the floor.
I have a problem with the wastes which are slow to drain.
I can live with the basin problem by not leaving the taps running.
The shower however fills up with water and can overflow.
If I use a drain unblocker (which is like a bike pump) I can make the water empty rapidly - once a vortex starts it is then OK.
I have noticed that when I plunge the sink that air comes out of the shower waste.
The waste pipes are connected only to the sink and basin.
To add confusion, when I first fitted the shower (about 4 months ago) it drained OK.
Any help appreciated.

Try one of the proprietary drain cleaners, one which states that it dissolves hair.
It's a good idea to pour lots of boiling water down first, as that softens soap residues, and quite often does a good clearing job.

Thanks for the reply.
If the pipe were bocked, with say hair, would it empty OK after plunging?

Thanks for the advice - tave tried the chemical attack - makes no difference.

Does anyone have another idea.

Thanks.