If Polk had not occupied Columbus (or, better yet for the South, had never been made an army officer), it is entirely possible that Union forces would have moved to occupy Paducah and other strategic points sometime in the fall of 1861. That makes the North, and not the South, the ones who violated Kentuckian neutrality and that makes a HUGE difference. About a third of the Kentucky population was made up of fence-sitters who, IOTL, were outraged when the South moved into Kentucky; ITTL, that anger would be directed towards the North.
People point out that there were more Kentucky units in service with the Union than the Confederacy and suggest that this is evidence that the state as a whole was overwhelmingly pro-Union. But this ignores that obvious fact that Kentucky was occupied by the Union for almost the entire war and that it was vastly easier for the North to recruit soldiers than it was for the South. It's impossible to know for sure, but my own estimate would be that the state was about 60%-40% in favor of the Union. If the fence-sitters had been made to throw in with the Confederacy in the fall of 1861, that proportion could easily be reversed.
It's also pointed out, correctly, that relatively few Kentucky men joined the Confederate army when it briefly occupied the bulk of Kentucky in the fall of 1862. But it's also worth pointing out that the Confederate Army encountered lots of goodwill throughout the state, with people coming forward with supplies, cheering crowds welcoming the Southern troops in many towns, and other displays of pro-Southern sentiment.
Had secession taken place, the whole situation in Kentucky would have transformed radically. It's worth remembering that Virginia had been generally pro-Union up to the moment of secession, but that Unionist sentiment essentially vanished (except in West Virginia) the moment that secession became an accomplished fact. Kentucky could easily have been much the same way.
As for what would have happened, I think Lincoln was correct when he said that to lose Kentucky was to lose the whole game. A fair chunk of the 75,000 men Kentucky sent to the Union Army during the war would instead be fighting in gray uniforms, the vast logistical and agricultural resources of Kentucky would have greatly added the Southern cause, and there would have been tremendous geographic advantages as well. Just as a single example, the Tennessee and Cumberland Rivers could have been defended much more easily on Kentucky soil than from Forts Henry and Donelson.