This applies even with perfect information, as long as drivers are completely selfish.
(Also, is this specified somewhere? It seems like the paradox relies on people making their minds up ahead of time (hence imperfect information) and then not changing their minds halfway.)
If you imagine that one of the Start->A->End travellers switches to Start->A->B->End, we now have 1749 cars travelling Start->A->End for 67.5 minutes, 501 cars travelling Start->A->B->End for 45.01 minutes and 1750 cars travelling Start->B->End for 67.51 minutes. The car that switched improved its own travel time from 67.5 minutes to 45.01 minutes, but the total time spent travelling increased to 258750.01 minutes (one driver saved themselves 22.49 minutes, but cost 2250 other drivers 0.01 minutes each).
So yes, if you found yourself in that second state, to improve the global optimum you would have to move to the first state by de-optimising the route of one of those 501 Start->A->B->End drivers.
The nash equilibrium would be for both prisoners to deflect but this is worse than if they both stayed silent.
Both staying silent is not an equilibrium because if one of them decides to be selfish and cheat, he can do better for himself.
Basically the information you need is agreement/cooperation ahead of time to know that neither side will cheat just to do slightly better for himself at the cost of others.