What slang words or phrases do British/American English speakers use for (poor) immigrants?
One of the ones I could think up of was offensive, so I wouldn't list it.
There is also "Pomegranate", which is rhyming slang with "Immigrant". This term is shortened to "pommy" and used by Australians to refer to Britons.
"Wetback" is a derogatory term for Mexican illegal "immigrants", not humorous and shouldn't be used.
I have heard the phrase "wretched refuse" used to obliquely refer to poor immigrants. It is a phrase lifted from the poem on the Statue of Liberty, instantly recognizable to any American with a small modicum of education. As such, it actually has a slightly more positive connotation than such a phrase would otherwise have. It probably would only work for immigrants to the USA though.
Generally a term like that is only really needed during an immigration wave. Such waves tend to be centered from one country, so every such term tends to be associated with a particular nationality (usually as an epithet). As such, you will have great trouble finding a word or phrase that doesn't double as a specific racial epithet.
An American term in wide use over a century ago was "wop," or "without papers."
It referred particularly to Italian immigrants, and there are other, less flattering references to them.
Can't think of any general BE terms - perhaps asylum seeker if you are the Daily Mail.
Generally it's just the name of the county abbreviated. There are a few offensive terms for inhabitants of certain countries, mick, frog, kraut etc but not specific to immigrants.
There was a case recently where a politician got into trouble for saying 'Jap' for Japanese, he claimed it was the official abbreviation for Japan but it was ruled to be offensive like Paki for pakistani.
More of a euphemism than a slang term, but my local public library has a couple of shelves designated as the 'New Americans Center'. (Why it's a 'center' I have no idea.) Of course, many people might resent being called an American before they wanted to be called an American, if that makes sense. On the other hand, I think it's a gesture that wouldn't necessarily get made a lot around here.
As someone has mentioned, FOB is definitely offensive, but I thought I would note that I've not heard it in the Midwest, only in California and almost exclusively to refer to Asians (often by other Asians).