I wouldn't worry about a single cherry pit, unless Tillie starts having symptoms. Elissa (my daughter) left a lot of cherry pits on the floor of my bedroom a while ago (long story - I wasn't happy with her at the time lol, because of Ceylon) and although I thought I cleaned them all up, every once in a while I find Ceylon chewing on a random cherry pit that he found when he's in my room. Of course, I make him spit it out every time I find him chewing on one, and I search again for more pits, but so far he has suffered no ill effects (and I am pretty sure that he has swallowed more than one).
Hydrogen peroxide is
the recommended method to get a dog to throw up, but the first, last, and only time I tried it (after Ceylon ate 3/4 of a chocolate brownie left in the car by my daughter), it didn't work (Ceylon was fine, of course - he just never threw up, at least not until hours later). Not sure if I didn't give him enough hydrogen peroxide or what, but I won't give it to him again until I talk with my vet about what to do if he does really eat something toxic, and the correct dose to give him. And hydrogen peroxide will only work (I have read) if your dog ingested the toxic thing within the last couple of hours. Again, I don't think you need to be worried about the cherry pit, but, being prepared in the case in which your furbaby does eat something truly toxic, is a good thing. I would suggest that you talk to your vet about what really might be toxic, and what you can do immediately to give your dog as an emetic if recommended (some poisons should NOT be thrown up), and have it on hand JIC.
But, although some dogs might react badly to a cherry pit, most dogs will pass it through harmlessly. If Tillie is having bad symptoms, I think that you would have noticed by now, and taken her to the vet already. Otherwise, I really wouldn't worry about one, or even a few, cherry pits. (Just my ignorant opinion, I am not qualified to give real canine medical advice, and I am sure some other more knowledgeable people on this forum will chime in