More than 140 members of parliament have signed a letter to the prime minister calling on her to resolve an anomaly that means many people who arrived in Britain as children between 1948 and 1971 are being denied health services, prevented from working and in some cases threatened with deportation.
There is growing anger that long-term British residents have fallen victim to rule changes in 2012 aimed at stopping overstaying. This meant that their legal status changed despite living, working and paying tax in Britain for decades.
Many have been told they need evidence including passports to continue working or getting health treatment. But many arrived on their parents' documentation and never formally applied for British citizenship or a passport.
"That these individuals are being treated with such contempt, disrespect and lack of dignity is a national disgrace," said David Lammy, an opposition Labour member of parliament and the author of the letter to the prime minister.
The immigrants are named after the Windrush, one of the first ships that brought Caribbean migrants to Britain in 1948 in the aftermath of World War Two, when labour shortages meant that people from the Commonwealth, a network of mostly former British colonies, were invited to help rebuild the economy.
Almost half a million people left their homes in the West Indies to live in Britain between 1948 and 1970, according to Britain's National Archives.
British media have reported cases such as a man who was denied treatment for cancer and a special needs teaching assistant who lost his job after being accused of being illegal immigrants despite living in Britain for more than 40 years.
May will hold talks on the issue with counterparts from Caribbean nations this week and she only became aware of a request for a meeting on Monday morning, her spokesman said.
"She is aware that many people are unlikely to have documents that are over 40 years old and she is clear that no one with the right to be here will be made to leave," the spokesman said.
A Home Office official said the rejection was because the subject of the meeting was not clear.
An online petition calling for an amnesty for those who arrived in Britain from Commonwealth countries in the Caribbean as children and a lowering of the level of documentary proof required from people who have lived here since they were children has now attracted more than 136,000 signatures.
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