Sounds very similar to a problem that me and a friend had when we bought a bungalow and rented it out.
We renovated it and replaced the bathroom, etc.
When the plumber removed the old toilet - it broke (same as yours!!).
When the tenant moved in, the toilet regularly got blocked, so we removed the toilet and used drain rods to try and unblock whatever was causing the blockage.
This didn't help, so we called out a professional drain service company. The guy put a camera down and had a look down the pipe all the way to where the pipe met the main sewer in the road. He didn't see it the first time, but after a few attempts he noticed something white lodged in one of the bends. It turned out a piece of the old toilet had fallen in when the plumber had removed the old toilet. The guy manged to dislodge the piece and push it into the main sewer - the sewer pipes are wider, so it wouldn't cause any more problem in there.
It sounds like the same has happened to your toilet - a piece of old toilet probably fell in when the plumber was removing it and it's now got stuck. Call in a professional drain service - they'll be able to sort it out for you.
Hope this helps.