You are right the compounds provoking this burning/tingling sensation is called allyl isothiocyanate.
We (human) perceive these compounds in two different ways when ingested, namely via the gustatory and olfactory systems. The molecular receptor sensing isothiocyanates is called the transient receptor channel A1 (TRPA1) [ref].
Here a simplistic view of how ingredients are perceived:
Gustatory sensation
First we get a burning sensation in the buccal cavity due to TRPA1 receptors present on the surface of the sensory neurons of the trigeminal nerve. In essence isothiocyanates dissolved in the saliva will activate those neurons via TRPA1 provoking electrical impulses in the trigeminal nerve which leads to a burning sensation.
Retronasal olfactory sensation
For several food ingredients we also perceived an aroma (taste = gustation, aroma = olfaction, taste + aroma = flavor). The aroma of an ingredient is perceived by olfactory sensory neurons in the nasal cavity and because some of the ingested ingredient will be vaporized inside the buccal cavity and travel through the nasal cavity where they are perceived (retronasal perception).
Not so surprisingly, isothiocyanates will again be perceived by sensory cells of the olfaction system via TRPA1 hence a burning sensation in the nostrils.
How receptor activation leads to a neuronal stimuli
This is a very broad subject and I will only give an overview of what is happening. We first start we the activation of a receptor by an agonist (in your question, isothiocyanates binding and activating TRPA1). Upon binding of the agonist the conformation of the receptor will change from an inactive to an active state. 762
The way the signal is transduced varies according to the type of receptor but for the TRPA1 channel what happens is that the channel opens leading to an influx of calcium ions ($Ca^{2+}$) in the cells which changes its electrical state (as calcium ions are positively charged), a phenomena called depolarization. Depolarization of specific sensory cells (here the one expressing TRPA1) start an electrical signal in the nerve which travels to the brain where it is then decoded. The exact way the brain decodes sensory signals is still under debates but a likely explanation is that only some neurons within the nerve are activated which allows the brain to decode the sensory signal based on the nerve activation pattern. As an analogy, it would be the same as having only specific wires (neurons) turned on within a bundle of wires (nerve).
Further reading
If you want to further extend your knowledge on taste perception at the molecular level read this excellent review <PERSON>, 2006. For a in-depth review on olfaction refer to this book. For a book on signal transduction refer to this while for TRP channels activation this is a good reading.
The field is quite broad and there are many good references. Sensory perception includes two distinct and broad fields namely neurosciences and cell signalling. 762