Most funding for Education PhDs comes from five sources: ESRC open competition awards, ESRC collaborative awards (awards part-funded by ESRC and part-funded by an organisation or company), project-based awards (funding that is tied to a specific project run by the supervisor), university scholarships, practice-based awards.
Unfortunately you have missed the deadline for ESRC funding to start in the next academic year (the deadline is in January / February each year). University scholarships usually work to the same deadline. Project-based awards will come up throughout the year, depending on when funding is received, but it's quite common for start-dates even for these awards to run on the academic year calendar because it ties in better with training provision.
To apply for an ESRC studentship you apply to a university that is part of one of the 14 Doctoral Training Partnerships (DTPs) and which has your Doctoral Training Pathway. To do this, you need to identify a potential supervisor or department based in a suitable university and follow the department's instructions for applying (usually you have to write a proposal and submit a CV. Some departments require you to do this before any potential supervisor is involved). Once you have been accepted, you then submit an application for nomination by your department / university for one of the ESRC Doctoral Training Pathways held by your university. If you are nominated, your application is then assessed by the DTP and they decide who should get funding.
The Doctoral Training Partnerships that cover London are:
- LISS (Kings, Queen Mary, Imperial) liss-dtp.ac.uk/studentships/studentship-competition-the-application-process/
- UBEL (UCL, SOAS, LSHTM, Birkbeck, East London) (Education Pathway is mainly UCL/IOE) ubel-dtp.ac.uk/esrc-studentships2/pathways/education/
- SeNSS (UEA, Goldsmiths, Reading, Roehampton, and Sussex have the Education Pathway) senss-dtp.ac.uk/apply
- LSE info.lse.ac.uk/current-students/phd-academy/esrc-doctoral-training-partnership (does not have an Education pathway)
The others are:
- Cambridge www.esrcdtp.group.cam.ac.uk/
- Grand Union (Oxford, Brunel) www.granduniondtp.ac.uk/education
- Midlands (Birmingham Loughborough, Nottingham, Warwick have the Education pathway) warwick.ac.uk/fac/cross_fac/mgsdtp/studentships/howtoapply/
- NINE (Durham, Newcastle and Queens Belfast have the Education Pathway) www.ninedtp.ac.uk/education/
- North West (Lancaster and Manchester have the Education Pathway) nwssdtp.ac.uk/how-to-apply/pathways/educational-research/
- SGSSS (Aberdeen, Dundee, Edinburgh, Glasgow, Stirling, Strathclyde) www.sgsss.ac.uk/studentships/
- South Coast (only Southampton has the Education Pathway) southcoastdtp.ac.uk/apply/
- SWDTP (Bath, Bristol, Exeter have the Education Pathway) www.swdtp.ac.uk/prospective-students-and-fellows/esrc-funded-studentships/
- Wales (only Cardiff for Education) walesdtp.ac.uk/studentships/
- White Rose (Hull, Leeds, Manchester Met, Sheffield, Sheffield Hallam, York have the Education Pathway, but Hull is psychology-focussed) wrdtp.ac.uk/studentships/
All their websites have information on applying for a PhD, applying for funding, what a PhD involves, etc. which are useful even if you aren't going to apply for ESRC funding. Some of them have ways to contact current students, ideas about how to find a supervisor.
If you propose to self-fund (or try for a University scholarship), you still have to apply in the same way (by identifying a supervisor / department and completing the relevant forms, etc.) but you can self-fund pretty much anywhere that will accept you and apply at any time.
Generally, Social Science PhDs are student-led. That means you decide on the topic, collect your own data and so on, with guidance from your supervisor(s). Some project-based and collaborative ones might be a bit more directed, but even then are usually quite open about what is studied within a broad theme.
A PhD must make a substantial original contribution to knowledge. It is better to have a supervisor who has knowledge of the subject of your PhD because they can help you to assess if your PhD will make such a contribution (you do not want to find out at the viva that other academics don't think it does!), as well as helping you with resources. For competitive places (and funding) that reject a lot of applicants, you will be unlikely to be offered a place unless there is someone there who is interested in your topic to supervise you.
It is difficult to assess whether you will get on with someone as a supervisor, although they will probably interview you before offering you a place which gives you some idea. Some departments will use a two supervisor model which has its pros and cons. Keep in mind that while supervisors should be pleasant and courteous, their job is to make sure you get a PhD, not to always be nice and keep you happy. There are times as a supervisor where you have to be honest about a student's work, no matter how hard it is for them to hear. Some departments will have profiles of their PhD students including the topics they are working on and their supervisors, so you can contact them to ask about their experiences. (e.g. Cambridge Doctoral students in Education: www.educ.cam.ac.uk/people/doctoralstudents/)
The word count can vary by subject but is usually around 70-100,000 words in the Social Sciences. If you are doing a full-time PhD, you are expected to treat it like a full-time job. Depending on the structure of your course, you may be expected to be at the university for 2 or 3 days a week in your first year to attend structured research methods training courses. After that generally it is up to you where you work, but it is good to try to be part of your department or university's PhD community simply for the support you can get from other students.
Personally, I find it a bit concerning that you do not know any PhD students in your own department. That suggests to me that either there are no PhD students or that any there are in the department are not well-integrated into the community and that they do no teaching. Neither is ideal in terms of the support you can expect there.