Thousands of people arrived in the United Kingdom as children in the first wave of Commonwealth immigration 70 years ago.
Guy Hewitt, the Barbados high commissioner, said: "We did make a request to the CHOGM summit team for a meeting to be held between the prime minister and the Commonwealth Caribbean heads of government who will be here for the CHOGM and regrettably they have advised us that that is not possible".
Named after the vessel that brought the first group of immigrants from the Caribbean to rebuilt post-war Britain in 1948, numerous Windrush generation arrived believing their rights were guaranteed for life, and never applied for a passport or other documentation confirming their residency rights.
And she said: "I do not want any of the Commonwealth citizens who are here legally to be impacted in the way they have".
Amid growing concern about the government's "hostile environment" strategy to counter illegal immigration, Ms Rudd revealed she was "concerned that the Home Office has become too concerned with policy and strategy, and sometimes lose sight of the individual".
Mr Lammy said: "What is going on is grotesque, immoral and inhumane".
The Immigration Minister added: "These are people who we welcomed here way back in the '50s and '60s and it's really important to me that we correct any error".
"This is about individuals".
"This is about individuals, and we have seen the individual stories, and they have been, some of them, awful to hear, and that is why I have acted", Ms Rudd said. "That is why I am so committed to ensuring that there is no cost involved".
Immigration Minister Caroline Nokes earlier appeared to suggest that some citizens may already have been deported by the Home Office amid confusion over their status.
Responding to a question on whether people had been deported in error, Nokes said: "Potentially they have been, and I'm very conscious that it's very much in error and that's an error I want to put right".
Br Ms Nokes could not give an estimate for the number of people affected.
On 15 April 2018, Nokes previously said in an online statement: 'I know that establishing status after so many years may be hard for some people but we will do everything we can to assist them.
This afternoon Ms Rudd said she was "not aware of any specific cases of a person being removed in these circumstances", instead telling MPs she was urging representatives of Commonwealth countries to come forward with their own examples.
Several cases have emerged of Windrush immigrants and their British-born children being denied medical care, losing jobs, or being threatened with deportation after being unable to prove they have a right to live in the UK. "And I would ask anybody, if they know of any such circumstances, they should bring them to the Home Office".
The PM's spokesman said the Home Office was expected to set out measures on Monday to support members of the Windrush generation in providing the documentation necessary to prove their right to live in the UK.
Mrs May's official spokesman said: "She deeply values the contribution made by these and all Commonwealth citizens who have made a life in the United Kingdom, and is making sure the Home Office is offering the correct solution for individual situations".
The Windrush migrants were advised that they need to produce evidence including passports, to continue working or receiving treatment from the NHS.