I can only explain how funding happens at UCL since this is where I'm located. Oxford and Cambridge will have their own specific sources of funding. All UK universities have this in common, though: PhD positions in Engineering are primarily funded via the Engineering and Physical Sciences Research Council (EPSRC). The EPSRC imposes very tight rules to ensure it spends its money only on locals:
- you must be able to stay indefinitely in the UK
- you must have continuously resided in the UK in the past 3 years
- you must have done so for purposes other than full-time education (though I'm unsure if that rule is actively enforced)
In rare cases, universities are allowed to spend some of their EPSRC money on foreigners. The Doctoral Training Centers I know about in UCL have one or two scholarships available to all.
Now getting into the specifics of UCL where I study. There are other types of funding available:
- UCL Graduate Resource Scheme: tuition fees + £15.5k for 3-4 years, but extremely competitive
- Discretionary departmental funding: depends on department, also covers tuition + stipend for 3-4 years
- Special scholarships: some scholarships are available on specific topics by scientific societies or for specific populations (e.g. women in computer science). You'll usually find a run-down of such opportunities on each department's application website as well as dedicated scholarship search sites.
- Lecturer/Professor money: case-by-case, but at least covers a £15.5k stipend as this is the minimum required for surviving in London. The money can come from industrial partners, FP7 programmes, EPSRC projet grants... This is probably the most common source of funding for non-UK students.
- Part-time research: that's not very common but totally possible, you can do your PhD part-time and have a job to support you. You'll still be expected to spend half of your time on your PhD so the job ought to be part-time too, and you should not attempt this under £16k a year post-tax (income tax and council tax). Being a TA does not pay well enough (£4-5k a year at most, if you teach a lot).
You should expect most sources of funding not to cover your tuition fees, which amount to roughly £4000 a year. In any case, in UK universities the application process is completely decoupled from the funding process, so you should apply and get accepted on academic merit before your proposed supervisor starts looking into funding, and the availability (or absence thereof) of funding does not affect the admission decision at all.