Should I consider legal action if my attempts to persuade my university to allow me to complete my PhD prove unsuccessful?
My broad diagnosis of your situation is that your time in your PhD program was too short and too long ago to give you a clear understanding of what the process of getting a PhD entails.  That lack of understanding is definitely going to cause you problems so should be addressed right away.  What other problems you may have, if any, is hard to tell at this point.  

Let me now try to address some specifics:  


  Nobody has told me that I cannot complete my PhD yet. The university has not responded yet, and I suspect they never will.


I'm already a bit lost.  The university will certainly respond to halfway reasonable questions directed to the right people.  Whom are you asking?  You should start by talking to people in your department: your advisor, the director/coordinator/chair/whatever of graduate studies, the department head....


  I am considering legal action. I'm wondering whether I shouldn't just force the institution's hand with (at least) the threat of legal action, rather than expecting someone else to fight my battles for me. At the same time, however, I don't want to start issuing threats. I would do anything to avoid legal action, to be frank.


Please read this over yourself.  It's conflicted bordering on contradictory.  But let me chime in: no, you should absolutely not try to "force the institution's hand" with the threat of legal action -- certainly not at this point, and in my opinion and experience, virtually never for someone in your broad situation.  The institution knows its own rules and policies intimately and you don't seem to know them at all.  The institution employs a team of lawyers; you do not.  A legal threat is not going to spook them into any positive action.

Also, IANAL and doubly not a lawyer in the UK, but in order to bring legal action you need to have some legal basis: a law they broke, a contract they breached, something like that.  You didn't mention any such thing.  Timing out your PhD may not be very nice, but that's not the same thing at all.


  If I were allowed to do so, I could likely complete in October 2018, so three years late.


You haven't been in academia for several years, but you think you could complete your PhD within three months from now?  Unless you have a completed thesis ready to go (or better, already submitted to your committee) that is basically impossible.  Your unrealistic time frame is a bit worrisome to me.


  I made a breakthrough in my second year, which would be the subject of my thesis. This work has been ratified by colleagues both within and outside of academia.


This is a key point in regard to how close you are to finishing your PhD.  "Ratified" is a strange word to use, and "outside of academia" is not very relevant either.  If your "breakthrough" resulted in a publication in a strong journal, then you're going in the right direction.  But you still need to work out with your thesis advisor whether the work of this publication alone is sufficient to be written up into a thesis (such writing may or may not take substantial extra time).  If you haven't spoken to your thesis advisor about this -- or, and here I am cringing as I type and hoping I am way off, if you don't have a thesis advisor -- then you are much less close to finishing than you think, and whatever acclaim you have gotten from people outside of your department is good rather than bad but not nearly so directly relevant as you seem to think.

Let me end where I began: please talk to your advisor or other sympathetic department personnel.  Please do not try to strong arm the institution at this time.  Good luck.