Unfortunately, that isn't going to happen. You can't turn up in the UK and then start looking for work, you have to have employment secured before you arrive. Also, cheap apartments don't exist in London, in terms of rent London is far more expensive than somewhere like NYC.
Moving in either direction between the USA and UK is pretty difficult. You could only relocate here if you were an independently wealthy investor, legitimately married to a UK citizen, transferred here by the arm of the US government for which you work or were a highly qualified and experienced doctor, engineer, scientist, chemist, teacher etc.
No disrespect intended, but we have no need for teenage shop workers, we have plenty of our own, so objectively it wouldn't benefit this country to allow you to move here.
Unless you have an EU passport as well as your American passport then your chance of getting a wotk visa is practically zero. As you're working at 19 I'd also assume you're not studying at university (college), most jobs require a mimum of an American Master's degree. Even the schemes that permit young people to work over here would not aplly yo you if you're not a university (college) graduate.
I'm afraid you will find it very difficult to obtain a visa to work.
London is one of the most expensive places in the world to live. There is no such thing as a cheap apartment.
Here is a link to Zoopla to give you an idea:
In my logical and experienced mind there is the concept of contacting the correct Embassy of the country anyone wishes to move to. The embassy staff has the proper applications and list of requirements for anyone that wish to enter the country to visit, to study, to work, to live, etc. Our world is very mobile and the United Kingdom welcomes qualified and unqualified workers each day. A few years ago in a visit to the city the hotel we stayed near the Tate Modern Museum had employees that spoke 15 different languages, the most fascination one was Swahili. Asking here is futile and a waste of time for no one here is a representative of an embassy of the UK.
It's not going to happen. There's no such thing as a cheap apartment in London. As a 19 year old American, you don't have the qualifications, skills or experience to obtain a visa. Sorry, but that's just how it is.
Not going to happen.
It's quite a thing, but almost all the questions posted here by seemingly young and unqualified individuals from the US wanting to emigrate to the UK are anonymous. Why is that? Maybe there is a bit of disingenuous posting going on, as they all seem a bit familiar.
To work in the UK as an American you need advanced educational and professional qualifications plus a job offer from a company that will sponsor you for a work visa. .
You can only move to the UK if you have the correct visa and work permit
So look into that first, by visiting the UKBA website
Remember unemployment in the UK is high for young unskilled people
The British drug store chain called Boots belongs to Walgreens. If they will give you a transfer to one of their British stores, it will be easy to obtain a work visa. If you work in one of our larger cities public transport is excellent. You do not need a car
in order to qualify for a visa you would need
a degree in chemistry, engineering or maths. shortages in these areas.
several million to invest in UK businesses
independently wealthy
sent by your employer
you do not qualify for a visa.
