How much can a Professor influence admittance of his/her PhD students? (UK) I was wondering how much influence a professor at a competitive university in the UK has in PhD admissions, when he/she really wants to work with somebody.

I am mentioning UK specifically, since the PhDs here are shorter and its maybe clearer what exactly the student will want to work on for his PhD.
To give a counterpoint to astronat's view, in Computer Science funding usually, but not always, comes (in one way or another) from a budget of the prospective advisor (e.g., her/his grants, startup package, etc.). So if (s)he really wants to work with you and funding is available (not all professors have funding to start with), you will likely be offered funding.  This is often formally independent of admission into the PhD school (i.e., a professor promising you funding does not formally guarantee admission), but in all places I am aware of the de facto rule is that candidates that professors wish to spend money on are also admitted as long as they fulfill the formal admission requirements.

Note that in many places in Europe the inverse is also true - candidates that no professor agrees to fund are often not admitted, independent of CV and letters.

So to answer your specific question:


  I was wondering how much influence a professor at a competitive university in the UK has in PhD admissions, when he/she really wants to work with somebody.


Assuming it's computer science, I would assume that the professor has a very large influence on admission.