A postscript to my earlier response .... I mentioned that many thousands of those Germans and Austrians who arrived in London were 'registered' at Bloomsbury House, including a large number of young women who then went into domestic service. The files created were lost after the war, but in the late 1980s a significant number, relating to about 35,000 individuals, were fortuitously rediscovered. These are now in the custody of World Jewish Relief (HQ in London) and have been digitally scanned and indexed. Amongst them was the file on my mother, Karola Grünebaum (DOB 12 May 1921, Offenbach), It is extremely interesting and revealing, not only including papers related to her arrival in 1939, but also to what happened to her in subsequent years up until 1943. WJR is keen to link the files with the descendants of the people who are documented, so it is well worth getting in touch to find out if your relative's file is one of those that has been found.
As others have mentioned, the National Archives (Kew, London) also may hold significant information, for example related to internment, naturalisation and the '1939 Register for England and Wales' (equivalent to a census).
Dr Sohail Husain, Hampshire, UK