Importance of supervisor support in being accepted for PhD this may seem like a silly question, as I know every institute is different. However, I was wondering how likely is acceptance to a PhD program when you have a supervisor on your side?
Background: I am applying for an incredible PhD in Ireland. I have very interested supervisors, who loved my proposal and I had two very promising meetings with them, one via Skype and one face to face. They offered me a desk and a new lab to set up, but they informed me it's now down to the academic council. They informed me it’s a formality and there is a high chance of acceptance unless something dramatic happens.
I will add I do have a weak undergraduate degree, but a strong masters which resulted in a publication. I also have worked as a lab technician in Italy and London so have a good bit of experience outside academia. 
I may be overthinking a bit as I am waiting for feedback from the college, and starting to feel the nerves.
Unless you fail to meet the minimum requirements for admission, having an advisor who is willing to support your application and commit the resources to support you as a graduate student is generally sufficient to get admission in Europe (and at many schools in the US). Most of the time, even under “normal” circumstances, the people responsible for admissions will want such a statement of support from someone in the program before an offer is given.