OK, going back to your OP:
First of all, let’s make sure that you understand the difference between a tornado watch and a tornado warning.
A tornado watch means that the National Weather Service believes that conditions are favorable for tornado-producing storms to develop in the watch area. Watches are usually issued for fairly large areas, and for extended periods of time (hours, at least).
A tornado warning means that a tornado (or, a likely tornado) has been spotted in the area, usually in one of two manners:
a) Actual visual confirmation
b) A radar return which is consistent with the formation of a tornado
Warnings are issued for much smaller areas (often, only one or two counties, and even then, perhaps only parts of those), and for shorter periods of time. A warning is not a “forecast”, it is a report that a tornado is actually occurring.
As I indicated earlier, there have been improvements in tornado forecasting; according to this report, those improvements led to an increase in the average “lead time” for a tornado warning, from 6 minutes to 11 minutes (from 1994 to 2002). The report does not give an estimate for a number of lives saved, but common sense says that, if you have, on average, an additional 5 minutes to react and reach shelter, there must be some benefit in reduced casualties.
It sounds like there are (at least) two separate questions in here:
- Does the presence of tornado sirens save lives?
- Does compliance with tornado warnings save lives?
I don’t know, to what extent, tornado sirens cover the U.S.; I would imagine that most areas which are not within earshot of a tornado siren are lightly populated, and / or rarely suffer tornadoes, anyway, so I’m not sure if such data exist.