There is very large uncertainty on this, and one should not attach much credibility to reports that do put a specific number on it.
Early work suggested that easy availability of real-time energy-use data led to quite high energy savings. Later work suggested that this had been significantly over-estimated.
Whether time-of-use pricing has any effect in and of itself would depend on the generation portfolio on the grid.
In Britain, gas plants are the marginal producer for an increasing proportion of the time (as coal plant gets decommissioned); moving demand around in time has little effect on carbon emissions when the only effect is to use one CCGT plant instead of another.
However, in the future, some method of time-shifting demand that's enabled or assisted by smart metering (maybe time-varying pricing, maybe something else) will enable a much higher combined penetration of exogenously-variable renewables such as solar and wind, possibly with some plant that would be too expensive to run flexibly, such as biomass with CCS.
So smart meters in and of themselves don't do much. It's the fancy demand time-shifting that offers the interesting opportunities.