I'll try and give an answer, but this is a tricky one for me to think about. :)
In my experience all three words are interchangeable and can be used either at the beginning of a sentence, or within a sentence between commas. All three can be used to indicate that you're introducing an exception to a rule. "However" can also be used to introduce a rule which is the opposite of the one you've just described and in that way it's similar to "conversely." Some examples:
In my social circle, people are expected to like to watch football. Nevertheless/Nonetheless/However, I do not.
In my social circle, people are expected to like to watch football. I, nevertheless/nonetheless/however, do not.
Noise and excitement are essential ingredients at a good sports event. However/conversely, silence and calm are necessary when studying.