I'm not sure I understand what prevents me from transmitting FTL. If I take it to the extreme and use very distant particles, what happens between when I measure one and the time at which light would reach one from the other?
The part you are missing is that the effect is very subtle. It only shows up in a statistical analysis after the fact. You can only notice/measure the effect if you repeat the experiment many times and compare the statistics of what is happening with particle B to the measurements made on particle A. If you look only at particle B then you have no clue about what was going on with particle A and if you only look at particle A then you have no clue about particle B. It is only when you combine information from particle B with information from particle A, after the fact, that you see any effect at all.
To learn more search for "EPR paradox". Basically what happens is that quantum mechanics violates locality without violating causality. This violates our common sense which is based on classical mechanics. Since wormholes are also non-local, connecting entangled particles with wormholes is more appealing to our common sense (for certain values of "common sense"). The interesting thing is that quantum mechanics already explains (predicts the outcomes of) entanglement experiments without any wormholes. So if you add wormholes (in any meaningful way) then you might need to change quantum mechanics. This is interesting because it might lead to a way of combining relativity with quantum mechanics which has been the unobtainable holy grail of theoretical physics for many years.
The chances of this working out are very very small. The chances of getting a Nobel prize if it does work out are very high.