The best estimate for modern dams is they will last for centuries before finally crumbling. The most likely failure mode for a modern concrete dam is actually the reinforcing steel, which will corrode in between 200 – 500 years without constant upkeep. Earthen dams, or ones made with a rubble core, don't have that limitation, but will likely also fail after several centuries of use without upkeep.
The issue for large dams over long periods of time is that eventually the basin behind the dam will silt up, and the volume of the reservoir will gradually decrease. If the flow of water into the reservoir remains constant, then the dam will eventually be overtopped and a waterfall will appear where the dam was, gradually eroding away the upper portion of the dam. There is also the issue of the pressures on the base of the dam; they were calculated for a particular weight of water, rather than silt (which is also being compacted over the centuries as more and more is deposited at the base of the dam.
For the sort of dam you are proposing, there is the additional issue of the dam being made of separate elements (rocks or bricks) which are held together by a combination of their mass and sealed with Roman cement. This sort of construction has thousands of distinct stress points, and the shifting pressure of the reservoir behind it over the centuries will gradually change the stresses on the structure in ways the designers may not have anticipated. As well, when the dam is overtopped, it will be far easier for the water to wash away a large structural element (one of the bricks or rocks), creating a channel which will amplify the flow of water and create a greater scouring force, eroding the cement and causing more elements to fail.
There are clearly a lot of different elements at play which make an exact calculation difficult. If the builder was in a hurry and didn't lay the courses of bricks or rocks properly, then the structure will be weak and may collapse with a high rain season or snow melt. If the cement wasn't allowed to cure properly, then you have another weakness. If the weather patterns change and the dam is called to hold back more water than it was designed for, then you have a pretty obvious problem. And of course, without regular inspections and maintenance by competent engineers, the dam will not last as long as it could.
So I would suggest that your dam will probably last for at least 200 years with no maintenance and excluding exceptional conditions like earthquakes or massive changes to the weather.