I'm assuming that you've already fallen out beyond repair with your supervisor? If not, employing a lawyer seems the fastest way to make that happen. Which if you are in the US will almost certainly have professional consequences for you, given the importance they seem to place on the letter of recommendation from your supervisor for job applications. I'd be careful not to torpedo your career in anger. And are you sure that there are no emails that you forgot to reply to, suggesting submitting the paper that your supervisor could produce as proof of assumed consent or anything like that?
Normally graduate students are quite pleased to get a publication. What's so bad about the paper being published in this place? Is the conference so poor that it's a minus rather than a plus for your cv? I'm not in a field that does conference proceeding but I know for comp sci for instance that a conference proceeding can be a good publication. Or have you found a bad error or something like that? If it's the latter getting your supervisor (given the professional embarrassment for him/her too) to negotiate replacing it with a corrected version might be a better path.

Better still might be to send a short document via email to the whole team before the meeting, with preliminary key findings and methodological challenges as headings. That gives you something to discuss in the meeting, you've prepared the ground that it's not complete, and your stats expert has chance to weigh in with advice at a stage where it might be genuinely helpful in saving you time, even if it's via email rather than in person.

Anz07 - oh no, I clicked hoping you'd finally had good news.
My advice would be to seek support from your supervisor if they are the supportive type. If they have been distant recently, it's probably because they know this but were forbidden to tell you. Ask if this situation has happened before and what happened. As well / Instead does your student union have any independent advisors? Or is there a departmental research student person or a good HoD you trust? Somebody who knows the university procedures inside out would be helpful to identify at this stage.
You need to know what should happen in a case like this, so that you challenge any procedural irregularities.A good supervisor would be doing this for you, which is why I'm suggesting talking to them, but you will by now have a good grasp on how likely this is. People will no doubt tell you to get a lawyer - I don't think this is that helpful at this stage as it would be expensive and finding one with the relevant expertise tricky. What might concentrate minds though is a few questions about the appeals and complaints process.
A similar case happened a few years ago where I work. In that case, a third examiner (a new external) was brought in to reevaluate the thesis and the degree duly awarded without another viva or any corrections.
I know you feel numb and you probably don't feel like a fight, but unless your supervisor is doing it for you, you do need to get informed and if you think they are not acting correctly, then start being everyone's prospective worst nightmare.

I was recently an internal examiner - it was an eye-opener to discover how crap our graduate school office is. Every stage of the process they sat on for at least three weeks. So even though I and the external had turned things round quickly the candidate still experienced delays at each stage and there was also a cock-up at this point.
Is there any way your supervisor (and by the way I think his/her initial claim that it would take a fortnight was really unrealistic) could find out from the internal where they are in the process? In the case I mentioned, it turned out that the external had returned all the paperwork but had overlooked signing one form. Rather than tell her, the office did nothing and it was only when the supervisor asked me gently about the delay that I knew there was an issue. Once I got involved the student had the result in 24 hours.

Agree that you need to talk this through with your husband - would he be ok with seeing little of his wife and children if you did live apart? Is there any flexibility with his job?
It sounds like the country you live in now makes it impossible to get a job in the short to medium term given the lack of childcare and the language worry. But Europe is a continent with lots of different childcare systems - is it feasible to get a postdoc somewhere where your childcare / language situation would be better? And would your and your husband's salary combined enable you to afford two homes? Do you have a sense of what the competition is like in Law to get a postdoc and how comparatively competitive you are? What about visas etc? I guess how practically feasible is the Europe plan?
Would moving back to your home country actually be better for you? I'm wondering whether family at hand might help on the childcare front, and obviously language would no longer be an issue. You don't really say what your objection to your home country is. It might be that the universities are less prestigious / jobs less well-paid but it's worth thinking about where you think your long-term future is and if that's your home country, then maybe starting to build a career there is the best strategy.

As you classify linguistics as humanities, I'm assuming you're at the modern languages end of the subject rather than the speech and language therapy end? If so, I'm afraid the outlook is not too promising regardless of how good you are. Nationally they think about 6% of PhD graduates will get permanent academic jobs. This is skewed by the much larger numbers in lab sciences but it's not much better in the humanities. If you are also trying to enter a field where UK student numbers are declining, which I think is true for Linguistics then that's an issue too, as govt policy looks to be to further cut non-UK student numbers, which has been the way out of falling UK interest for some programmes. I know for example that there are very high numbers of Chinese students on our linguistics programmes. Brexit and the likely impact on research funding is also likely to cut the number of entry level research jobs. And this is all in addition to the casualisation issue you mention, which is particularly endemic in the humanities. Basically it's a depressing outlook for universities at the moment. I think if not getting an academic job would be a deal-breaker for you, then the odds are probably not good enough to risk it.

You are making a very similar choice to the one I made nearly a decade ago. I was 5 years into a civil service job having gone in as a fast streamer and progressing reasonably well. I jumped to do a PhD and was one of the very few for whom an academic job did materialise, but I am less certain that I would make the same choice again if I was making it today. Things I think it's worth thinking about:

1) Your earning potential in the civil service is almost certainly higher than in academia even if you are very lucky on the academic track. Would never being a home-owner be an issue for you for instance? Unless you have a partner on a high salary, academic wages will not allow you to buy in much of the South of England these days. Civil service pensions and benefits are vastly superior to academic ones too so it really is a big financial hit to take. When I jumped the differences were less stark than they are now.
2) You presumably know that they think only about 6% of PhD grads will get a permanent academic job. Sadly the Oxbridge name is no guarantee. You're currently presumably earning a graduate salary now - it's not just the time to PhD but the history PhDs who made it into academia seem then to have spent an average of 4-5 years on temporary, part-time and poorly paid work to get a strong enough publication list to have a chance. It's one of the more overcrowded subjects. Again big financial hit to swallow especially if you'd like a family.
3) The civil service is a good employer much better than universities. Would your skill set offer a route back in (mine did which mitigated the risk a bit)? I ask that as many humanities/soc sci PhDs see the govt as an ideal second choice after academia.

Given your field, is it a possibility that you've fallen foul of any new guidance issued after the Stubblefield case? I know the journal(s) that published that work got slated for publishing the facilitated communication work she did (regardless of the criminal charges she was found guilty of), so I wondered whether the relevant professional association has put new ethics guidance in place, that a cautious editor might think would impact on your article. Journal editors are responsible for checking the ethics of the research complies with disciplinary guidelines, even if not noted by referees, as they're the ones who carry the can if they publish something dodgy.

Not sure there are any, as different universities use different categories of outcome, but if there are HESA would likely be the place to look.

It's worth having a chat with your supervisor about what disciplinary norms are for your subject e.g. on length of review, and to get a sense of how the particular journal is viewed in the field quality-wise. You might for instance be minded to recommend accept for a graduate journal that you know struggles for submissions, when you'd recommend reject for the same article at a top-ranked journal.
Don't reject something because the article doesn't use your preferred theory or method - try to be open-minded however committed you feel to your own research choices.
I saw this on twitter and thought it worth posting as even if nowhere near your subject might be helpful for someone doing a search in future:

Two thoughts.
1) The financial outlook given the HE Bill, Brexit and probable new limits on international students is poor throughout the UK HE sector. Cuts are the new norm, so you might be experiencing this wherever you were.
2) if your supervisor is noticeably more research active that her colleagues, then she's probably trying to publish her way into a more research-oriented post. That means she has a vested interest in your success in publishing etc - that could work in your favour more than being one of a very large crowd of PhD students in a stronger dept.

I don't think it really would help. And it carries the risk of being viewed by employers as a failed PhD. Have you talked to anyone about funding? My gut feeling is that with a 1st at undergrad you might be writing this off too hurriedly. If you are geographically mobile and could find suitable supervision at several of the less competitive AHRC doctoral training partnerships, it might be a few applications next Spring.

There are few project-based funded soc sci PhDs - most of the funded places available are through the ESRC's doctoral training centres, where you have to write a proposal. If you meet the eligibility criteria, and can put together both a really strong proposal and a good supervisory team then, it's not impossible for someone with your profile to get funding particularly if you aim at the DTCs based away from London / Oxford / Cambridge.
Regarding self-funding - you need to ask yourself whether you would be happy to have spent all that money (and add in lost earnings during the PhD) if as is likely you don't get an academic post at the end. Nothing to do with your ability, just that the stats say that only 6% of PhD graduates in the UK will get a permanent academic post. If you are wealthy then maybe that's irrelevant, but if you are going to struggle to cover fees / living expenses (and thus really struggle to pay too for conferences / fieldwork), then if it didn't work out, it can be gutting. I understand the LSE is trying to cut back on self-funded PhDs for this reason.

The 1+3 PhDs are often linked to research council funding doctoral training centres, and do tend to start in September. Only the universities the research councils think are strong enough in their research training for a particular subject have DTCs. The DTCs tend to have one deadline a year usually early Spring. Other fund PhDs are tied to specific research projects and can be advertised and start at any time.