Hmm... I thought the conductor uniform being stolen was so the watch could be placed in the room for whatever reason, but it's possible it could be KID I guess...
Anyway, thanks to Lynz, Manga and Yuri for trying to help me with this

. Tried to take into account what you guys were saying and hopefully I didn't fail too much

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So... trying to prove Fusae is Mystery Lady. Attempt 1

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Loose ends from Fusae’s first case
There are quite a few things not tied up yet from the first case Fusae appears in. The main one is, of course, Billy saying that Fusae and Agasa will meet again way before 10 years pass. This means he has some intention or knowledge of when this will occur and it's probably for some other reason than just meeting up again. From the sounds of things, he knows they'll be nearby and Fusae can meet up with Agasa again then, though I'm trying not to read into it too much. Add to that DC's Comic Book Time and you can be fairly sure it'll happen before the end, maybe only a few months in-universe.
The next point is the three meals in the car when the DBs discover it. However, only Fusae and Billy are about and neither of them seem like big eaters. Normally, Aoyama is very obvious about which characters are like that by how they look. Agasa and Genta are that subtle really. Plus, Aoyama wouldn't just add this in for nothing. Both of these are clear points that they'll be reintroduced at some point.
Also, Fusae's heritage is made a big point of and I think will be relevant further on. It's a big part of her character that we've seen. She's essentially based her life's work off Agasa's comment so someone being accepting of her heritage must be pretty important to her if it's made that big an impact.
I have to wonder if her suddenly leaving the school and her having a stepdad will also be important later. After all, the time when she was growing up divorces were very rare so it could be her dad (who I'm presuming is the on on the Japanese side of the family because of Billy's name and stepdad status) has died. However, with interracial marriages at that time, it could just be they felt too pressured to continue. Personally, I prefer the idea her dad has died because of the possibilities it opens up, but I'm not decided yet.
Link to Akai
This part is mostly a collection of things I've read and stuff I've added on. A vague outline about this can be found
here from Chekhov.
Random stuff to back it up:
-It works well with Okiya-Akai theory. If Akai has a link to her then suddenly Okiya does too, it can back up the Okiya-Akai theory by a different perspective.
-If she's unlikely to be recognised because of how long ago the file is, it makes it easier to pretend she's Vermouth in disguise to readers who don't remember her.
-Going by the manga, the timing perfectly matches up.
-Fusae has some foreign heritage. Going by how Aoyama usually works, this will either be American or English and most people who live/visit there will end up meeting at some point according to what he's done so far.
-The loose ends from her case need tying up, and the shadow people spying on the DB and Yukiko that case still haven't been explained either.
I don't think she's FBI, due to the fact that Akai and Conan seem to be trying to fool them atm. She doesn't have any links the BO know about to Akai or he wouldn't use her. Her stepdad is probably involved too somehow, so it could be something family related, however as it seems SA(and so, most likely, the BO too) know about Masumi, I'm not sure if that would work.
Timeline of events
I really don't think I can do better than AJM about this so I'll link back to him/her again. Their theory also contains some more info about their possible link to Akai. Try
here, but the necklace thing is probably just a fluke.
VAs
Fusae’s VA is
Eiko Masuyama, who’s also voiced Fujiko from Lupin III, and is a veteran VA. It is likely Aoyama specifically requested her and I get the feeling she'd be more expensive than most due to her veteran status so just getting her in for a love plot of a side character like Agasa doesn't match up.
Billy, who Fusae appears with when meeting Agasa is also voiced by a Lupin III VA,
Kobayashi Kiyoshi, who voices Jigen. This means the same applies to him as it does to Fusae and if Fusae reappears it's extremely likely Billy will be nearby.
It seems like Aoyama's playing around Lupin III references, though I really hope it doesn't get as far as with the Gundam stuff.
Why not Vermouth
My main arguments for the Mystery Lady not being Vermouth are that the woman is with Okiya (who is most likely Akai), and that SA is most likely Vermouth during the current case. With Mystery Lady and SA being seen at the same time it's impossible for both to be Vermouth.
Link to Okiya = Akai theory is
here.
My attempt at explaining why I think the current SA is Vermouth is quoted below and I’ve edited some bits to cover what I missed before.
Stopwatch wrote:
SA can now talk convincingly enough like Akai to convince Sera s/he's the real thing and considering Akai is Sera's sister, I don't think she'd be fooled easily. That means someone with experience in disguises. Aoyama wouldn't chuck someone in out of the blue so the only choices are KID (unlikely as plot and KID don't mix and his case is in a month's time anyway), Yukiko (why would she disguise as him? It'd just put her in unneeded danger, though it is possible) and Vermouth (fits in with Amuro being undisguised and not being shocked at someone else being disguised as Akai. This SA has to be someone he'd recognise, prob'ly org and Vermouth is the only member who fits the description)
Vermouth told Gin/Vodka that Sherry was aboard. As far we know Amuro(who I'm presuming is Bourbon) didn't see Haibara and anyway, he doesn't know Haibara is Sherry as far as I can tell. SA both saw Haibara and knows the Sherry-Haibara link so s/he is the only available option I can see to tell Gin/Vodka.
Why not Kir
Kir is under heavy surveillance by the BO. I doubt they'd let her out of their sight without having her watched. Also, Gin and Vodka were aware that Bourbon and Sherry would be aboard the train before it set off so even in the very unlikely case they let her go off without being watched, they wouldn't let her anywhere near such a delicate operation, especially as Gin has a personal stake in this and doesn't want anything to go wrong. Basically, Gin won't let someone he doesn't fully trust, but has no evidence against, near to Sherry, where they can mess things up.
Besides, Kir can't take unnecessary risks at this point without losing any hope of defeating the BO, so being in contact with Okiya(Akai) could get them both killed and ruin the entire point of faking Akai's death in the first place.
Why not Yukiko
I'll start with the weakest thing. In file 821, SA doesn't pay any attention to Mystery Lady until
after she's passed by. If Mystery Lady was Yukiko it's likely SA would've shown at least some interest. Aoyama had a lot of frames on that page where it could've been shown. My point with this is that with SA most likely being Vermouth, wouldn't she recognise her friend? They've known each other for years and yet, a hat prevents her from recognising her? That only works out of universe because of Aoyama's art style. Seems unlikely it'd happen much in-universe now (Kaito/Shinichi and Ran/Aoko excluded of course because they're intentionally similar).
Also, what would Yukiko's link be to Akai? Conan is too private to let anyone in on his speculation (Hattori, Agasa and Haibara in the Vermouth arc and the FBI when faking Akai's death if you go for that theory). Add to that his protective streak when it comes to people he knows being in danger and you can pretty much guarantee he won't tell them in order to keep them 'safe'. This behaviour can be relied on because, face it, he'd've been better off telling Haibara his plan in the Vermouth arc, but didn't in order to 'protect her' and look how well that turned out.
The only other way Yukiko could become connected to Okiya is by arriving back at her house and seeing him there. The thing is, Okiya has had random people coming across him in this house plenty of times and, sticking to his story about the arson case and Conan/Shinichi letting him live there is what he always does. For Okiya-Akai theorists, the fact Okiya(Akai) is a trained FBI agent will help a lot too. Yukiko might be curious as to why Conan would do that, but he won't talk and she won't find out from Okiya.
You'll notice I've not mentioned Yuusaku yet. That's because not only is he based on Mycroft which means he won't get involved in anything, but him figuring out about Okiya and everything then telling Yukiko and lurking during the case kills quite a bit of the suspense. If there's all the plot going on and then Yuusaku pops up to fix and explain everything, it'll ruin the story.
There's also the fact that when they first found out that Conan had been shrunk and was possibly being hunted by the people who had already tried to kill him, as soon as Conan said "Nope, this is my case" they backed off.
The only way I can see either of them being involved in some way is if Fusae is linked with Interpol somehow which is unlikely at best.
Last point on them, Yuusaku and Yukiko are like KID and Hattori in that they only appear every so often. They've had their very recent appearance and returning again so soon breaks the current pattern.
Why Aoyama didn't forget and it wasn't too long ago
We know from the most recent file that Aoyama hasn’t forgotten about hints as far back as Volume 41 when Elena’s tapes first appeared. Fusae first appears only a volume before that, not long before the final hints and climax of the Vermouth arc. This time was when the plot was heating up (or as much as it can in Conan) and lots of little hints are everywhere at this point so it makes sense that Fusae can be linked to the plot, even if not evident at first. It's similar to how Araide was portrayed pre-Vermouth arc climax, though there aren't as many hints.