There have been a few attempts, though none got anywhere. Afaict, there are two main problems with tagging these timings:
Measuring them in the first place
Some traffic lights are adaptive, so don’t have any kind of fixed timings. Others vary their timings over the course of a day - think rush hour, off-peak, night time. Measuring any individual timing at some point in time might not be accurate at other times, since signal timings are usually determined on a case-by-case basis.
Verifying them
Let’s say you did all that work, carefully measured all timings, entered them, etc. How does someone else verify that it’s actually correct (and not out of date, which can also quickly happen with timings)?
Basically, they have to do all of that themselves too. I don’t think that any city anywhere publishes those timings, so survey it is. Thus, nobody wants to maintain it.
Apart from that is the issue that traffic lights themselves don’t have a tagging scheme detailed enough to even ascribe attributes to each of them. That would need to happen first before timings can be added.
This only makes sense in comparison to the green duration. If the traffic light is 10s red and 10s green, that’s basically the same as 5m red and 5m green. A router doesn’t know signal timing, so both cases have a 50% chance of red and green when you arrive (not considering the yellow time to simplify).
This is basically the conditional probability version of the other case (If I got green at this traffic light, what is the probability that the other one is red or green?). This is a bit simpler because it doesn’t require absolute information.
Some further thoughts
The usefulness for routers is somewhat limited by the fact that they can’t predict the future. Most of the time, they aren’t that wrong because the problem space is pretty much continuous (start 1 minute later => reach the end 1 minute later). Traffic lights are discontinuities - start 1 minute later, reach the end ??? minutes later.
This is incidentally also true for public transport stops (start 1 minute later => potentially miss your bus/train/flight). We can still work with it, because every single “signal timing” (stop duration) is written down on the time table. Traffic signal timings are basically time tables. With every single traffic light being a stop.