If they eat the leaves or flowers you can see vomiting. I would expect to see signs within 12 -24 hours if at all.The bulbs are more toxic and can cause vomiting and diarrhea, possibly with blood.There are no effects to the kidney (lilies are members of the same family, and can cause serious renal damage).
I'm not sure where you got that information, but tulips only cause GI upset, varying from mild to quite severe depending on the portion ingested as well as the total quantity.
I few websites listed GI upset, drooling, CNS depression and seizures as possible problems but seemed to indicate that the bulb was the most dangerous part of the plant which he was not exposed too. If it is just GI upset, I am not as concerned. He is a pretty big cat 18 pounds but tall and long. I also read you could put caster oil in milk to soothe upset stomach but that seems as though it could add to the problems. Is there anything else I should do at this point of just keep an eye on him>
I recommend simply watching him. If he does exhibit GI signs, he can be treated with anti-emetics at your regular vet. If any GI bleeding occurs, he would also be given oral GI protectants and antacids.