If you over-stay your allotted time by a day or so because you missed you flight you will probably not get into trouble and as you say it might be fun for a while to feel like an outlaw. When I visited the United States I certainly felt tempted to do just that. I didn_t though.
In the US you are obliged to carry identification with you at all times and if you are stopped by the police for any reason you will most likely be caught.
I believe they won_t go out of the way to look for you but if they find you, you will be arrested, tried, punished and deported. You will then find yourself in trouble with the UK authorities as well as the US authorities and you will probably find it impossible to ever return to the US.
There are bigger problems though. For example, how will you earn money, without identification? In the US there is no free medical care so how would you obtain and pay for medical insurance? You will need to obtain false credentials and that would get you into even more serious trouble.
I really don_t think it_s worth the risk over a bit of fun.
They would not look for you or know you haven_t left. you would just officially be asked to leave if you came into contact with border control, etc. - which would only happen if you tried to travel internationally.
You would still be able to use your credit and debit cards from the UK.
But you_d have to work for someone for cash without documents (millions of central americans do it).
I_m not sure if you could open a bank account without a work number. But perhaps you could have a bank that is also located in the UK and use it here.
As long as you were doing something productive and not trying to live off of public services I don_t see any big harm.
They would flag your passport, which would make it very difficult to travel within the US. They_d also flag your credit/debit cards, so you_d have no access to cash. You would be sought (although I doubt they_d put you on the FBI_s most wanted list unless you share a name with a terrorist), and when they found you, you_d be deported and banned from ever re-entering the country. You might also get a criminal record.
It might sound like fun, but it isn_t worth it...
I really don_t know.
I assume someone would report you missing though.