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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.
No professor can reasonably be forced to take a PhD student they don't want. So there are only two sides/aspects to this question.
Surely, for (1) the decision is all with the professor, but not with (2). There are various reasons for why this will not work out, such as
A professor may have a small influence on the admission of a candidate, but the way admissions decisions are made will vary across departments.
I think, in general, if an applicant is already known to a member of the department and they would be happy to take them on as student, that applicant's chance of an offer of a PhD place is increased. However, I would still expect that a formal application, satisfactory references and an interview with other faculty members would be required before an offer was made concrete.
I don't know what field you are in, but in mine (physics), funding for a PhD generally comes either from a research council (i.e. the government) or from the university itself. A single professor will have almost no influence on where that money goes. Funding decisions may be made after admission decisions and completely separately to that process, unless the candidates are applying for a specific funded project.
Speaking from my own experience, I was made an informal offer of a place at a university where I know one of the professors very well. However, funding in that department was scarce and the money ultimately went to another, stronger candidate (as decided by a committee that my supervisor had no influence on). Consequently, I interviewed elsewhere, and was fortunate to be made a fully funded offer at another university.
To summarise, if a professor really wants you on board, they may be able to offer you admission, but the funding decision is likely to be made by someone else. (This of course changes if the professor has some grant money of their own by which they can fund a student, or if the student is self-funded.)
