I get a sewrage stench in my bathroom only early in the morning. Plumber cannot solve the problem.

I know you called a plumber but could be the seal is broken around the bottom of your toilet and the odor is seeping up from there. Is it an older toilet? Are you on septic? Are you in a trailer? More information would be helpful for a good answer.

Tick

Got kids or an old husband? Just kidding, most odors are sewer gas entering via a bad trap or wax ring under the toilet. Tell us what that plumber did when searching for an issue. That will speed this along.

The plumber changed the trap with a "Vent on trap"and he changed the cap of the outside vent with a 2-way-cap and lenthened it to the roof.We sealed off the overflows of the bath and the basin and keep in the plugs and keep standing water in it.the toilet has got a wax seal.Nothing helped.
The outlets of the basin and the bath goes directly into the main drain. What about making a separate gulley for them before goeing into the main drain?

You are on a septic tank, correct? This "two way cap" was a vent off the septic tank, right?
Looks like you covered the bathroom pretty well. Time to look else where. Did the plumber look up in the attic for a open or leaky vent?
I get a sewrage stench in my bathroom only early in the morning. Did you know that sewer gas will raise during the heat of the day and descends in the cool of the evening? This could be why the smell's worse in the morning. All we can do with smells and noises is point you in the right direction. I would check the attic and if all else failed I would have my plumber put my house under a smoke test to see if any of the joints are leaking.
. What about making a separate gulley for them before goeing into the main drain?Please explain what you mean? Regards, Tom