You mentioned the neurotransmitters released at the synaptic cleft, but seem to be interested in the complete picture. An action potential is actually very slow to propagate down an axon with a conduction velocity from 0.5 to 150 m/s, depending on how myelinated the axon is (https://www.ncbi.nlm.nih.gov/books/NBK10921/). A toddler can run faster than the slowest signals! The length of axons in the brain varies greatly as well, from a few micrometers to a few centimeters, but only a fraction of a percent actually grow longer than one centimeter.
We will make a very rough calculation. Pretending the average axon length is half a centimeter (5 millimeters), and the average conduction velocity is 25 meters/second (or 25,000 millimeters/second), then the delay incurred from the action potential down the axon will be 0.005 seconds (or 5 milliseconds).
This doesn't include the time it takes the presynaptic cell to release neurotransmitters, the postsynaptic cell to absorb the neurotransmitters and transduce them into electrical impulse again, and propagate the electrical impulse to the soma for integration. For that, the delay is somewhere around 0.5 to 4 milliseconds (https://www.ncbi.nlm.nih.gov/pubmed/14980200).
Obviously there are lots of variations in all these numbers. So how long does it take for a signal to be transmitted from the soma to the neuron's axon terminal to another? A few milliseconds. How long does it take for release and absorption of neurotransmitters? Perhaps another few milliseconds. If you're the type who wants a simple, idealized, singular number that ignores the complexity of the facts, then I would estimate something like 7 milliseconds for the total process.