I am facing the same problem as you, pretty much identically. I have now left Oxbridge off my CV - just covering that info with "BA (Hons.) in [subject]" and not mentioning other qualifications at all. I'm guessing you work at a City law firm, as do I. My feeling is if someone knows about the legal field, they will know that there is a certain level of education needed beyond university - and if they don't, then they're not going to think about that when they look at your CV.
One one application form, I had to put my salary, which as I'm now looking for work well outside London, can look excessive. I was told (after they had appointed a candidate for the role I had applied for) that my salary expectations would be too high - as it was a public sector role, I knew what the salary range was and I still applied! No point arguing the case with that sort of rationale. I've not mentioned my salary in other applications and when it's come up in interviews, I'm able to give very good reasons for not needing the London wage any longer. If only those others had thought to ask me, rather than making an assumption on my behalf!
I think a lot of people are applying for the same administrative roles as me, but they possess the exact experience needed. As a recruiter myself, if I were hiring for an admin role, I would be looking for someone with direct admin experience, not a solicitor who probably has a PA! (No offence 
!)
The approach I have now taken has been to get personal referrals from contacts who know people in organisations I am interested in working for (one of the benefits of the old OB education is you're likely not to be shy about networking). I have written in this semi-speculative way to three now and today had an interview for a role that was not advertised, plus am meeting informally with another to discuss freelance work. So this seems to be paying off.
Is this something you can do - even on the most speculative level, find out the name of an HR person or Fundraising manager and give them a call to ask if they would mind you sending them an email as you're massively enthusiastic about working for their organisation?
Out of interest, are you looking for part time or full time work? I've found that part time jobs tend to be even more difficult to convince people that you genuinely want to work there - can't put my finger on why.
On the HR side, you might find it hard to get taken seriously without some sort of HR qualification - even the very junior HR staff at my firm have taken some sort of course as it's a highly regulated field and they like to see that people are keen to keep up to date with that sort of thing. If you can afford it and have the time, might be worth looking into the CIPD courses on offer.