She isn't trying to hide, unless simply being seen would cause someone to ask about her ID? She's just trying to survive in a world that's very different from her own. I could, perhaps, have her start as a cleaner at the pub, then she sees how things are done, they get short a worker on a busy night, so she offers to help, and they promote her. Is that feasible?
What Shakesbear said re cleaners being on in the morning, but also, she is from 700 years in the future. If you were presented with the average cleaning kit of the 1311 house, how would you get on with it? Is she going to know about basic stuff like wringing a mop out before you slop the water onto the floor? How about plugging in a vacuum cleaner?
As it stands now, she finds a couple of students working their way through university and rents a room at their place. They accept cash and don't ask for references. Later, this comes back to haunt her, because someone comes around investigating and so they kick her out on the street, realizing she might be illegal.
When I was a student 10 years ago (cringe), it was the standard in a student house that the rooms were rented individually from the landlord. In a student house, each bedroom usually has a lock on the door to prevent this situation (students subletting the house and not telling the landlord). Also, the students are unlikely to take her in because of the council tax issue (I say unlikely, but it depends on your students, if they are able to sublet the place and they don't tell anybody she is there, they'd get away with it, but it's a pretty unlikely thing unless they were desperate for some money instantly, but even if they were, there are better ways to get hold of it, hardship loans etc. I'm not saying it's impossible, I'm just saying you run the risk of having things being oddly convenient. Especially as they later kick her out for being illegal - it's nothing to do with them, her being illegal.). Plus, she'd need to pay her bond and rent in advance. Will she have that money?
Nobody ever checks up on references so it would be perfectly fine for her to provide a fake one.
She has no clue how to use the internet. And she's not looking for real ID, she needs false ID, since she's from another century. Going to a government office and trying to get a real passport is going to raise red flags all over the place, I would imagine.
Her main problem would be that, according to them, she wouldn't exist. I don't know what would happen in that case. They couldn't necessarily deport her because they couldn't prove she was American either. She might end up in an asylum centre, but equally she might just end up somewhere safe until somebody can work out what her rights are.
I could be remembering wrong, but I could swear that I was told she'd need a passport even for a cash-under-the-table job. That the boss would want something to look at, so if they got busted later, they could at least claim they'd looked at her ID and didn't realize it was fake. But if she really doesn't need any ID to get a job, then I can take that part out.
Not a chance. I have never, ever had to show ID either for cash-in-hand or "proper" work. In the UK, you don't have to have an ID (and there was much fuss in recent years over efforts to make having one compulsory). If you don't have a passport or a Driving licence, it doesn't matter. They don't need to check if she can work because if she can, she'll have an NI. If they don't want an NI, they probably won't care if she's legal as she's sporting an American accent.
This is all very confusing for me, since I've never tried to get a job in a foreign country without ID. I wouldn't have the first clue how to go about getting a fake ID, or how to ask for an illegal job, or any of this. I just need her to be able to survive for a few months, until she catches the attention of London's Extraterrestrial Home Office. They'll give her refugee status, a real passport, a job, a place to live, and everything she needs. But until they notice her, she's on her own.
Which would make her vulnerable to getting picked up by a gang master who would probably sell her into prostitution, especially if she's naively going around trying to get hold of a fake ID.
As for getting a job, as said upthread, looking for signs in newsagents' windows, shop windows. Plenty of people advertise cleaners positions or ironing positions in those places and, to be honest, if she turns up looking presentable, she'll probably get the job (unless she's going to be left in the house alone to do it, then they'd run some checks).
I could just have them find her living on the street a few weeks after she time-travels, but then they'd be rescuing her, and I'm trying to avoid the whole "damsel in distress" element. I want her to survive on her own for awhile, to prove that she's capable of making it without their help, to show that she's got some character, that she's strong and clever, rather than a meek princess waiting to be rescued.
It's difficult to suggest things because I don't know where she's from or what it's like there. If she's pretty and clever and confident, you could have her doing some light grifting - http://en.wikipedia.org/wiki/List_of_confidence_tricks#Other_confidence_tricks_and_scams - scroll down to other confidence tricks and schemes. Stuff like Empty Car Lot would work. Or the Stranded Traveler Scam. This would get her some money to solve the immediate problems such as having the readies for a bed for the night. At the moment, I don't even know if she's familiar with basic stuff like how to use a telephone.