I've been haunting this forum for a while, got some real help from you, thank you all

But... I'd like to ask if anyone knows how long it takes to process the passport.
Our situation: My partner applied for a fiancee visa in August 2018, it got initially refused (because their info was out of date about an NHS bill that was paid well before we applied). We appealed, and the decision was overturned just before it went to the tribunal. We finally got the email for her to send in her passport, 27th March. She sent it in as instructed (To VFS in New York) and they got it on the 5th April.
Then, after chasing them to get a confirmation of receipt for a few days, they finally told us that as our application was so old, the passport had to be sent to the UK to get the visa. The change of process that allowed VFS to process the passport didn't apply to us, apparently.
So, it's been a month. I've chased VFS, but they say to contact UKVI in Sheffield. I've tried contacting them, but their paid contact lines can't look at individual applications (which makes me wonder what we're paying for...). Ive even contacted my MP, but his office is still waiting for a response. I've complained, but that can take up to 20 days for a reply.
She's getting really stressed because she's in temporary accommodation (Airbnb/motels) and we have no idea how long she will have to do that, how long to book ahead (going fortnight by fortnight at the moment)

Do any of you wonderful helpful people know how long it's likely to take to get her passport back?
On a secondary note - regarding their mistake on the initial refusal, is it worth going to the parliamentary and health ombudsman to see if we can get some of our expenses (due to the delay over their mistake) back?