I had one home birth (which went wrong, and ended up with ventouse then forceps), and one ELCS. Both involved cannulas. They are truly horrible, and very painful.
However... My own particular phobia is vomiting. I am absolutely paranoid about vomiting. So much so that I chose a drug-free home birth. The community MW had meptid as a back-up, but I rejected it at the outset, as I was so terrified of the thought of meptid (which has vomiting as a side effect, albeit an uncommon one. Even if it was one woman in 100,000,000 that risk was too much for me).
In the end, I was transferred to hospital and had every intervention under the sun, because DC1 got stuck.
There's every chance you won't need a cannula at all. But what I'm trying to say is that when it comes to it, all you will worry about is not your phobia. All you will want is for your baby to be born alive. You really won't care how that happens, should it get to that point. When it did get to that point, for me, my mum had to give consent because I was unconscious.