They're all clumsy phrasings, but for what it's worth I'd grade them (from most to least acceptable) as...
There was not enough commitment on my part or John's.
There was not enough commitment on mine or John's part.
There was not enough commitment on my or John's part.
There was not enough commitment on my part nor John's.
There was not enough commitment on mine nor John's part.
We can throw out the mine versions on grammatical/idiomatic grounds by noting that Google Books has 292 instances of "on your or our part", but only 3 of "on yours or our part" (my/mine = your/yours). And the nor versions are at least "unusual", in that nor usually follows neither, rather than not.
But however you arrange them, those last 5 words are just too "busy", trying to combine an inclusive "or" and two possessives (one a pronoun) associated with the same "part". It would be far better to avoid "problematic" constructions like this completely, and rephase as, say,...
There was not enough commitment from either myself or John.
For reasons I can't easily put my finger on, I personally would identify John first, and myself second. Perhaps because the last-named person "stands out" more, and it's "bad form" to put more of the blame on someone else in such contexts. Or maybe just because I'm influenced by the fact that The Queen often says My husband and I...
There was not enough commitment from either John or myself.