First, the new version of the British Library website describes its finding aids for Persian manuscripts clearly but briefly. I will just highlight a few of these and tell you how I have found them useful.
Of the basic finding aids for Persian manuscripts, some are aimed specifically at the mss. held in the British Library, while some attempt to address every Persian manuscript known to humankind. Of the general catalogues, the one you will probably want to start with is C. A. Storey's Persian literature: A Bio-bibliographical Study. Storey's volumes break down as follows (the volume/fascicle numbers change depending on how it is catalogued/which edition you're using):
- Qur’ānic literature
- Biography (corrections and additions)
- General history, the prophets of early Islam
- Special histories of Persia, central Asia and the remaining parts of the world except India
- History of India
- Mathematics, Weights and measures, Astronomy and astrology, Geography
- Medicine, Encyclopedias and miscellanies, Arts and crafts, Science, Occult arts
- Lexicography, Grammar, Prosody and poetics
I started using the Storey volume on astronomical manuscripts first, but obviously, for historians of India, the volume that contains the History of India is crucial. Also obviously, the volumes are somewhat out of date, even though you'll find plenty to keep you busy. Ghulām Ḥusayn Tasbīḥī wrote a doctoral thesis at the University of London on the problem of updating the volumes, so if you're wondering what you're missing, and why you're missing it, you can read "The Problems of Bringing Storey’s Persian Literature Up to Date (Persian Lexicography)."
Exclusively tuned to the Mughal Empire is D. N. Marshall's Mughals in India: a Bibliographical Survey of Manuscripts. This is good addition to Storey. One difficulty with this index (for me) is that the manuscripts are listed alphabetically according to author--if you know the name of a manuscript, but not the author, you have to do a bit of work to find the description.
If you're working on a science-related topic like me (astronomy), you might want to sit down with A. Rahman's Science and technology in medieval India : a bibliography of source materials in Sanskrit, Arabic and Persian (New Delhi : Indian National Science Academy, 1982). This is a great source, as it lets you sift through Persian, Sanskrit and Arabic science manuscripts simultaneoulsy. I should warn you, the abbreviation "I.O." does not stand for "India Office" in this volume. One strong aspect of this catalogue is the attention given to manuscripts held in collections in India--"I.O." refers to one of these, so if you try to order up a ms. numbered, say, "I.O. 4566," it won't work at the British Library (this I know from personal experience).
In the list provided on the British Library webpage cited above, you will see a few catalogues focused on collections now held in the British Library (previously held in the India Office, the British Museum, or elsewhere).
At the end of the nineteenth century, Carl Hermann Ethe began to catalogue Persian manuscripts in the India Office. The first of two volumes was published in 1903, with the second volume following in 1937. So, if you're trying to find a number and description for a Persian manuscript you think was probably originally held in the India Office (for instance, if your shelfmark/citation begins with "IO"), start with Ethe's Catalogue of Persian Manuscripts in the Library of the India Office.
Similarly, Charles Pierre Henri Rieu catalogued the Persian manuscripts held by the British Museum (along with Arabic and Turkish manuscripts). If you think the manuscript was held by the British Museum (shelfmark will begin with BM in older catalogues), turn to Rieu's 4 volume Catalogue of the Persian Manuscripts (1879-95) and his Supplement to the Catalogue of Persian manuscripts in the British Museum (1895) to read the description.
I won't give advice on how to order up I.O. or B.M. manuscripts using the British Library's Integrated Catalogue (online). Generally, you just use the numeric digits, but the process is impossible to describe in text. Ask the reference librarian for help--they can input the citation properly (even they sometimes have to try twice or thrice to get the right string of letters and numbers), then order up the manuscript for you by proxy. They do this all the time, so go ahead and ask for help.
Finally, since I am providing sources for historians of India, I should also mention Elliot and Dowson's The History of India: the Muhammadan Period, as Told by Its Own Historians (31 vols.), even though it is not really a catalogue. This is a very large collection of manuscript descriptions and translations of ms. excerpts into English. You can often read a short extract in The History of India, cross-reference it with Rieu or Ethe shelfmarks, and come up with the original Persian manuscripts 70 minutes later.
Oh, I promised to discuss the trouble-making "D. P." shelfmark. Storey cited a manuscript held in the India Office as "D. P. 627," noting that Ethe had not included it in his catalogue. I asked for assistance from two different reference librarians, but neither one could figure out the shelfmark. Finally, I was able to contact one of the curators of Persian manuscripts, and she told me that it was "Delhi Persian" papers. When I finally managed to get my hands on this manuscript, the staff at the Deliveries desk were mystified--apparently they only infreqently (never?) see the D. P. shelfmark. Storey described these papers in a separate listing, apparently not published. I'll let you know more about it later, as the curator is going to let me read Storey's description of D. P. 627 Thursday.