Is your niece starting Y10 or Y11? Was this the result of early entry at the end of Y10? If so, what English is she studying this year? A lot of schools enter people for English Language in Y10, then if they do badly they can retake it, or otherwise they can pack English Literature GCSE into a year. If she is going to take GCSE Language at Higher Tier plus GCSE Literature (only available at Higher Tier, I think, but I could be wrong) then that should be manageable, given support from the school.
"She wants to be a language teacher and has high grades across all other subjects including A* in French."
You do have to ask what the hell the school was doing putting her in for foundation English. Either the decision was wrong, or something had gone horribly wrong with the teaching and assessment in the school. But I'm confused by what year she's in: has she got an A* in French GCSE at the end of Y10? Impressive.
If by "language teacher" you mean "degree in MFL followed by a PGCE" then a C at English is a potential problem, and might rule her out of contention at some universities. It's hard to judge because it's three years before she'll be applying, and applications for MFL are way down and therefore it's not hugely competitive, but a C in English can be a problem. Someone wanting to apply for medicine (I realise that is not necessarily comparable) would massively restrict the medical schools they could apply to, for example. And more generally, most serious applicants to Oxbridge and the more competitive Russell Group universities will have mostly As and A*s in their best eight; a C will stick out like a sore thumb.
It's also possible that between now and five years' time, when your niece applies for a PGCE, the floor maths and English GCSE standard could be raised for PGCE applicants. I wouldn't say it's likely, but I wouldn't bet heavily against it either.
In my city, A or A* in what you want to study, plus B in maths and English if they aren't amongst your A Level choices, is the basic entry requirement into the more academic sixth forms. There are other sixth form colleges that will accept you with C in maths and English, but they will still want B in the subjects you intend to study. English, French and German (or Spanish) is the standard A level suite for MFL degrees, and very few places will let you study A Level English with a C, still less off the foundation paper. So she really does need to retake it, I'm afraid.
"Should her previous coursework be allowed to be submitted?"
Either it can or it can't, and there's nothing the school can do about it. Find out the facts, by talking to the examinations officer at the school or the exam board in question, and accept the answer.
" Should it be a letter to the HT asking for how this happened "
Suppose they say "OMG we totally messed up, slapped botties all around". How does this help? You are where you are.
"Should it be raised with the examination board separately?"
Should what? She entered a paper with a C-grade cap, and got a C-grade. How's that the board's problem?
"Or should she just suck it up despite the complete lack of empathy and support?"
If she decides she needs to retake, complaining about the school's failings is a luxury she doesn't have. Hopefully the school will be supportive. Don't start arguments that aren't worth winning about what happened before: focus on the objective, not the history.
"My sister is not overly academic and does not know what to do to help"
But she's asking the right questions. I'd say, not knowing more of the facts, that a C at GCSE from the foundation English paper was not likely to be helpful to someone who wanted to study MFL at university, and all other things being equal it would be worth re-taking.