In microbiology (my field) there are quite a few studentships available which cover a stipend, tuition fees and bench fees, therefore you aren't required to have funding.. Alot of these are funded by the BBSRC and you just need to fulfill their candidate requirements i.e be an EU student, have a 2i in a biological science and not be a mass murderer (more info on requirements on bbsrc.ac.uk)
One thing with these studentships is that the project has already been predefined and so you will be working on a set piece of work which is slots into the work of the research group.