If there are trees, rocks, etc.... then humans WILL ALWAYS have tools to work with. So, the advantage that the vegans have over the omnivores in terms of lack of tools and weapons is basically moot.
I believe that the vegan population would dwindle much more quickly then the omnivore population. Not every one in the vegan camp would die from malnutrition but enough would pass away that they couldn't sustain their own colony. Now if this hypothetical island was overrun by soy beans then they might have a fighting chance....
Omnivores have the additional advantage in being able to fish, hunt for clams and other shellfish from said fresh water as well as the ocean.
The omnivores will survive given the higher intake of protein along with the other additional food groups present on their respective islands. Better chance to build up a minor society and possible fleet of boats to escape said island.
Worst case scenario, the omnivores still will have a far greater intake of protein then the vegans. Enough of an advantage that they can invade the vegan's island and pretty much take the vegans ... lunch money from them without a fight. Remember, it's a dog eat dog world.
Languages wouldn't likely evolve that quickly that each group couldn't understand each other in the period of 10 years separated from each other.