The British Library has perhaps the most comprehensive collection of primary source material on India.
Copac, British merged catalogues database, which includes such likely suspects as SOAS.
British Library's new Main Catalogue, supposed to have an additional six million more records than the old Integrated Catalogue (19 vs 13 million),
Unfortunately, the British Library does not lend out books – they have to be read in the library itself.
If you are travelling there, here is how to apply for a Reader Pass: http://www.bl.uk/reshelp/inrrooms/stp/register/stpregister.html .
You may prefer to inquire about the possibility of having a copy made: http://www.bl.uk/reshelp/inrrooms/stp/copy/rrcopy.html (this page also contains links to information about British copyright law).
Here is their Customer Help Centre: http://www.bl.uk/reshelp/atyourdesk/docsupply/help/index.html
H/T Edward Proctor, Duke University library, for compiling this information.
In addition, the Official Publications of India project resulted in the deposit of more than 23,000 volumes of duplicate volumes from the British Library at the University of Chicago. The Official Publications of India collection includes the pre-1947 publications of the Government of India and of other governments within the subcontinent.. You can search the material online at the U of C library website but still you cannot request Inter Library Loan. You have to travel there to access the material physically.
Also at BL.., "The India Office Records are the documentary archives of the administration in London of the pre-1947 government of India. The 14 kilometres of shelves of volumes, files and boxes of papers, together with 70,000 volumes of official publications and 105,000 manuscript and printed maps, comprise the archives of the East India Company (1600-1858), of the Board of Control or Board of Commissioners for the Affairs of India (1784-1858), of the India Office (1858-1947), of the Burma Office (1937-1948), and of a number of British agencies overseas which were officially linked with one or other of the four main bodies. " Here is the catalog for their "India Office Select materials."
More archives in the U.K.