It sounds like your sewer is backing up:( Or maybe blocked by tree roots.
Smells come from lack of water in traps. Toilets, and sink, as well as showers,tubs all have traps. That is one source of smells that often plague people.
Is the water still sitting in the bowl after it flushes?
The 2nd thing that can cause this odor is a failed wax ring. This seal is placed between the bottom of the bowl where it sits on the floor and the top of the pipe that carries the waste away. If the toilet rocks even a little bit, this can compromise this seal flatten the wax and allow odors to come into the room from below the bowl. The worse part is it can allow just enough wet to exit the pipe and begin to destroy the floor from around the pipe to beyond the toilet.
Place your knee against the side of the toilet bowl, push into the toilet slightly, does it move? If so you will need to remove the toilet and replace the wax ring. Not a hard job, but just a bit tricky for the first time doing it. If the floor appears to be rotting, make sure it is repaired to allow for the toilet to sit tight. Replace the bolts also. These come with most wax rings.
The next thing that can cause odors is a broken air vent within the plumbing pipe system. Although rare, it can happen.
Do you have any access to below this toilet? If so that is the first place to check. If leaks, faulty wax rings or damaged vents have occurred this is the first place you will see it.
Are you on city water or well? Some wells have a bit of sulfur in them. This can make a room smell like rotten eggs. There are filters for this and perhaps you have one that has failed. Something else to check.
Thanks for your reply. The wax ring is good & there is water in the bowl.
I assume you have a sink in that room? Do you have an S trap or a standard trap in it. S traps have a tendency to fail at keeping water in them if the vent system is partly plugged. If you have an S trap you can check this, run the water in the sink for one min. Then turn the water off. Wait about five min. Then flush the toilet and listen at the sink. Do you hear any glub glub sounds? If so when the toilet is being flushed the suction effect of the flush is drawing air from the S trap causing what little water that is in it to be pulled out. This results in the sewer gas to be allowed to come into the room.
The last thing that can be an issue is how the sink and toilet is vented and its location to other lavs in the home.
Lynne even though your toilet doesn't leak (which would lead you to
believe the wax ring was good) it can still be allowing sewer gas in.
Two things that I can think of that would cause this is pouring hot
water down your toilet, or if the toilet has stopped up at one time
(usually it has leaked around the bottom, but that's caused from it
being stopped up and it blowing the ring) thing is once the clog issue
is taken care of they don't always continue to leak ( it doesn't leak
because of the law of "path of least resistance" lol or because it's the
type of wax ring that also has a boot) but even though it isn't leaking
water anymore sewer gas will still leak out. Oh and just so you don't
think I "googled" or that I'm just blowing smoke this is based on over
30 years in the only profession I've had, plumbing, the last 30 have
been at a University :) Hope this helps!