I'm a third year student currently, and I have a pretty clear idea of what I want to research and that I want to do a PhD. My university gives very little information on how to apply for PhD funding, so I'm nervous about the entire thing. Unfortunately, because of family commitments and no car etc I have to stay at my current university, otherwise I would be applying for studentships across the country. Can anyone give advice on how to approach lecturers with a research proposal? Or whether I should simply enquire first and then work on a proposal together using the resources they have access to. Thank you!

It's very rare, in the UK, for a lecturer to have immediate access to PhD funding - i.e. be able to decide to take you on and make a funded studentship happen. What normally happens with any grants they secure, is they go out to jobs.ac.uk at the point the project they're related to starts. Candidates then apply in a normal job application scenario.

Unfortunately it's a bit of a false assumption that Universities fund PhDs (though most have small-scale, fought over schemes); the majority of funded PhDs come from UKRI or EU projects whose investigators are at the institution. This means there are comparatively few rolling 'open' calls for applications. If your Uni doesn't have an immediately apparent scheme, it's possible one doesn't exist.

It's very likely you'd need to move for a funded studentship given this, since you'd need to be applying for studentships funded by projects in your field, and the chances of there being one starting at the perfect time, at your institution, and you being the best candidate, are pretty slim.

Universities are, of course, open to self-funded applications. I think the reality might be you need to decide between mobility, or, if it's absolutely not an option and you're determined to do one, doing a self-funded full or part-time one at your local institution.

PS - if you go the self-funded route, it's still worth asking if the Uni could meet you in the middle, if you're on track for a good degree, and have a supervisor that's read your proposal and is interested in it. They will be unlikely to fully stipend you, but you may be able to get a fees waiver. It's usually considerably easier for an interested supervisor to use what little power they have to get the university to waive the fees in exchange for a really good potential of a timely PhD completion, rather than pay out a stipend.

abababa is right. The most common funded PhD route is applying to an already funded PhD with a fixed topic/supervisor. If you are limited to your current university a research masters might be useful as a stop gap.