As @radicalOH mentions, there are several timescales depending on whether you are talking about immediate, short-term, or long-term synaptic potentiation, each coordinated by a cascade of events (that have been extensively documented in you're interested in knowing more).

To specifically address your question about how quickly learning happens in neurons... it's fast. Is it ~10ms? mmm, that seems incredibly fast, and I'm not sure I can provide any definitive answer. I think it heavily depends on what you're willing to interpret as 'learning' when examining events on the molecular level. There are events that engender near-immediate alterations to ongoing transmission; and other events that support lasting electrical changes (necessarily?) develop across longer timescales. You might think of this distinction as the difference between being able to hold a phone number in consciousness via repeating it over and over, and then actually being able to recall that phone number 5 minutes later. In the former certainly something internally must have changed because a moment ago you weren't repeating this number in your head, and now you are. On the other hand, can we really say it was learning if the number is immediately lost the moment it leaves consciousness. Anyway, I"m not going to wax philosophical on that. I'll just show you some data for timescales we think are sufficient to produce lasting increases in synaptic efficacies...

Here is a video showing a single spine just after 2-photon 'glutamate uncaging':

https://goo.gl/rfgjrW

(as I mentioned above, glutamate receptors are the primary receptors involved in fast-excitatory transmission; so uncaging tons of glutamate right next to a spine is going to evoke immediate plasticity changes - particularly under these experimental conditions)

Note that in just a few seconds the spine has doubled in size. Morphological change in the spine are probably not the first events that alter synaptic transmission, but morphological changes to the spine are a reliable sign of a lasting memory trace.

Here is a whole cell recording of the electrical response in a neuron to incoming signals, before and after stimulation like that above ('EPSC' stands for 'excitatory post synaptic currents'):

https://i.imgur.com/0OTMwH5.png

There are multiple time scales synaptic plasticity happens on. For instance, ion channels can be moved in or out of the membrane to modulate excitability. On a longer time scale, gene expression can change. Synaptic plasticity can get really complicated, but Kandel's book has some great chapters on mechanism.