The only real con to a natural 3rd stage is that you may or may not hemorrhage. Risk of heavy bleeding is really the only possible negative. To my knowledge here is really no telling if you will or will not have heavy bleeding after delivery. Yes, it may take some time to naturally deliver the placenta and often OB's are not OK with this as they like things to be within their time frame/on their schedule.
I'm not sure how things are in your area, but at all the hospital births I have attended it is hospital procedure to routinely give Pitocin after delivery of the baby. This facilitates the uterus clamping down (causes strong uterine contractions just like when used to induce labor), may minimize heavy bleeding, and speeds up the delivery of the placenta. Cons, it's not what nature intended and artificially speeds up the process, it may cause intense cramping. Immediate skin to skin with the baby and baby breastfeeding within the 1st hour will naturally to the same to release oxytocin and contract the uterus back down. Pros, if you are bleeding heavily the Pitocin may minimize the need for transfusions and in extreme circumstances could be life saving I suppose. If you are planning on having and IV or receiving an epidural during labor (which requires an IV) the nurse will hang a bag of Pitocin before delivery and run the whole bag wide open after baby is born (I don't remember how much it is, but it's a lot!). If you deliver without an IV, you will receive and IM injection in the thigh after deliver (cons of this are it takes longer to get into the blood stream). That is most likely the shot you were speaking of.
If you are planning to delay cord clamping you will have to speak to your OB about what "delayed" means to them (it may not be as long as you would like) and make sure you are specific and bring it up again at your birth before delivery and as soon as your baby is born. The only con I know of is that it may lead to higher instances of jaundice, which in my opinion is not a big enough reason not to delay clamping. It is also very common for OB's to practice "cord traction" and tug and pull on the cord as the placenta is close to delivery. Cons, the placenta can prematurely detach from the uterine wall causing hemmorhage or retained pieces of placenta, the cord can break.
If you are wanting to avoid all of this and go the natural route, the best way to do this is to discuss your wishes with your care provider, really speak up for yourself, hire a doula to help advocate for your wishes and, avoid IV and epidural. In my personal experience clients have really had to fight the no Pitocin after delivery/ natural 3rd stage issue in a hospital OB setting.
This is just my doula advice, knowledge, and experience!