Funding for PhD students in the UK for EU student If I wanted to do my PhD in the UK at a renowned university in the natural sciences, how likely is it that one gets access to a fully funded PhD program?( living costs + fees).
The reason why I am asking this is, that I see a lot of scholarships on the websites of the universities but I have never heard of an actual percentage of PhD student(coming from the EU) who got such a scholarship. Does anybody know? It does not have to be an actual number, but maybe you know something about this.
I do not know actual numbers unfortunately but I will tell you what I know from my experiences in a Computer Science and a Statistics department (of different universities). Your question is tricky because there are a lot of different cases:


BBSRC, NERC or EPSRC funded places (by far the most common). UK Research councils are the standard funding bodies for PhD programs in UK; they usually do not cover maintenance fees unless you are a UK national. They pay only the registration fees for EU students. (I know, I am one). The success rate for something like this is the highest compared to the following options; it generally equates that of a PhD program acceptance rate to the university you are applying to. 
University/departmental funded places. Generically also called PhD bursaries. UK universities have self-funded places but they are not very common; usually not more than 2-3 places per department. In most cases those are money on hand but registration fees are occasionally lifted, I think because you are an EU national that might be easier (I know that for an Indian national fees where not waived.) their acceptance rate is obviously harder than those of option 1. 
Industry (or goverment agency) funded places. Industry collaborations do occur, I have seen some brochures but I don't know anyone that got them. Pretty much the same situation in terms of fee coverage as option 2 but they tend to be even more rare. I would guess it is not "harder to get" than usual departmental funded places given you actually apply to one. Almost certainly these are tighten with a faculty member and they are not "up for grabs" by all applicants.
General university scholarships. These are usually merit scholarships and you compete against the whole university. They are hard to get: In the medium size university I am now this translates into 25 places. They do not differentiate between costs, they give you a lamp of money (about £13k-15k) and you are on your own. It is not completely unusual to get one if you are an EU national and you have already a registration-fee-only scholarship. Obviously these are really hard to get, but selection is non-random: if you are great you might get them.
Supervisor money. Usually they are tighten to specific grants/research programs/etc. It is almost the same as option 3. This is weird money for you to directly tap into. Essentially they are money a supervisor/department has and it is decided that they'll be used to fund a PhD. They might not even be advertised. I know two cases, in first one fees were waived, in the second one not. Success rate? I have no idea, probably the probability of really impressing with your research potential, a successful and well-connected professor with a shortage of good PhD students (Given you are good, the hard part is actually finding the professor; not many people have a couple tens of thousands of pounds to allocate to a person they think is awesome.) 
EU programs. Sci-fi. I know only post-docs that got these kind of funding. I can not give any estimates, I am just mentioning them cause I know  they exist. I have come across them in relation with the Erasmus Mundus program, worth checking.


From what I understand, if you are an EU national registration fees are somewhat easy to be waived (ie. paid by the university or a funding body). Getting the extra money is the hard part but if your supervisor wants you, strings can be probably pulled. Teaching also pays good in UK in comparison with other countries. In general, do not let finance hold you back from applying to places. Especially for an EU national if they believe in you they will find a way to make you a somewhat good offer as your registration fees will not be horribly high to start with. Good luck!