Short answer: Your first sentence isn't correct at all. You can use either of the conjunctive adverbs to connect the two clauses, but they shouldn't be there together. Also, some believe it's better to use a semicolon to connect two complete sentences.
Long answer: Nevertheless is a conjunctive adverb:
A conjunctive adverb is an adverb that connects two independent clauses. Conjunctive adverbs show cause and effect, sequence, contrast, comparison, or other relationships. (Emphasis mine)
This option is fully correct:
I would like to start, nevertheless, it is not possible.
About although, it's categorized as a subordinate conjunction. So this is correct also:
Although I would like to start, it is not possible.
Note that connecting the two sentences using both nevertheless and although isn't correct as both are some kinda "conjunctions"!
But wait...
This is rather pedantic, but some people (including the author of this tutorial) believe that if you are to connect two whole sentences, a semicolon is preferred over a comma.
So, the fully correct sentence with nevertheless is:
I would like to start ; nevertheless, it is not possible.