It all depends on individual cops and individual situations.
One cop can walk past your sleeping form and another will run you in.
Once they find out that you’re not a UK citizen ( and so not eligible for legal aid ) it’s quite likely that you may find yourself up before The Beak in the morning. Obviously, once it is established that you have no means of accommodation, this will mean a strong chance of deportation. If that happens it will, inevitably, bring you to the attention of the US Embassy as they will be asked to fund the deportation of someone travelling on a US passport ( we don’t deport people free of charge ).
As I say it will all depend upon what mood a police officer is in, but if it’s discovered that you have enough funds to travel to this country in the first place but have not provided sufficient funds for accommodation or made any arrangements for accommodation, it could go a little harsh on you.