What is the timeline for broken plumbing to cause damage to a house's foundation?
The answer to your question is that there is no answer - there are just so many variables that it's really impossible to answer with anything quantitative.

The biggest variable is how much water is going down the drain.  If you had a clog but very little water usage, it could be days before you saw anything on your basement floor. If you have a family of 8 taking showers every day it will show a lot quicker.

The smells start being noticable when sewage starts backing up on your floor, until then its unlikely you'd notice anything.  Again, this can vary.  If you are just putting fresh water down your drain, it might never smell.  My neighbour didn't notice until they had several feet of water in their basement!

The rate of deteroiration will depend on age and condition of foundation beforehand, temperature, soil conditions, what the house was built on, quality of construction, thickness of foundation, etc.