The first proven European scholar who had the idea that Indian mathematics might have anything of note for the Western scientific tradition was Charles Matthew Whish.
The first person in modern times to realise that the mathematicians of Kerala had anticipated some of the results of the Europeans on the calculus by nearly 300 years was Charles Whish in 1835. Whish's publication in the Transactions of the Royal Asiatic Society of Great Britain and Ireland was essentially unnoticed by historians of mathematics. Only 100 years later in the 1940s did historians of mathematics look in detail at the works of Kerala's mathematicians and find that the remarkable claims made by Whish were essentially true. [From here]
Whish's article, 'On the Hindú Quadrature of the Circle, and the Infinite Series of the Proportion of the Circumference to the Diameter Exhibited in the Four Sástras, the Tantra Sangraham, Yucti Bháshá, Carana Padhati, and Sadratnamála', is fully accessible. There seems to be confusion on whether this was fully written and published during his lifetime, with the Royal Asiatic Society's Transactions placing it into December 1832.
Whish died in 1833, and his brother deposited C.M. Whish's considerable collection of palm-leaf scripts to the Royal Asiatic Society. These seem to have been his primary source, and have been catalogued since then. Regarding these palm-leaf scripts, the Royal Asiatic Society says:
While most are from the eighteenth and nineteenth centuries, a few are much older, with our oldest palm leaf manuscript dating back to the 12th or 13th century AD. ... Whish was fascinated by Vedic philosophy and classical literature, the study of grammar and philology, and the history of mathematics and astronomy, and the collection reflects his interests.
Of Indian mathematical treatises and mathematicians, Whish's article noted the following in detail:
- Sankara Varma (Sadratnamala)
- Somayaji (Charana Padhati)
- Talakulattara nambudiri (Tanta Sangraha)
- Cellalura nambudri (Yutki Bhasha)
The following other texts were also mentioned:
- Surya Sid'dhanta;
To note the style of the work, as several comments on the OP noted, this was often related to commentaries on preceding work:
The testimonies as to the author, and the period in which he lived, are the following, viz. The general consent of the learned in Malabar; the date which is shewn in the commencement of the work itself, namely, the year 4600 of the Caliyuga ; the mention made of him in the first chapter of a work named Driccaranam by his commentator, the author of the Yucti-Bhashu, CELLALURA NAMBUTIRI... This is the evidence of the author of the Yucti-Bhashu, the commentary on the Tantra-Sangraha, concerning the author of the latter work: the date of the Driccaranam is mentioned in the latter part of the work, viz. the 783d of the Malabar era; and in the summary account of the periods of astronomy, it is written 4708 of the Caliyuga, both of which coincide with the year 1608 of the Christian era.
Whish's article was not particularly impactful over the next century:
Whish's paper was not completely forgotten over the following 100 years - it would be more fair to say that it was largely ignored. The first paper that really carried on Whish's work was the 1944 paper. This paper by K Mukunda Marar and C T Rajagopal begins:-
This paper is a sequel to an article bearing the same title contributed more that a hundred years ago in the Transactions of the Royal Asiatic Society of Great Britain and Ireland, by Charles M Whish of the Hon. East India Company's Civil Service in the Madras Establishment. The article of Whish has come to be accepted as one of our chief sources of information concerning Hindu achievements in "circle squaring", but the questions it raises with regard to the date of these achievements have still to be answered. ...
As mentioned above, once Rajagopal and Marar took up the baton, they managed to popularize the premise of very successful pre-European Indian mathematics. Cadambathur Tiruvenkatacharlu Rajagopal is described as:
A final topic to interest him was the history of medieval Indian mathematics. He showed that the series for arctan x discovered by Gregory and those for sin x and cos x discovered by Newton were known to the Hindus 150 years earlier. He identified the Hindu mathematician Madhava as the first discoverer of these series.
I couldn't find Rajagopal's article's original, but the University of St. Andrews, Scotland, overview quotes the following relevant passage:
A little over a century has elapsed since the first attempt was made to mark on the map of modern scholarship this virgin continent [Hindu mathematics]. The person who sighted the unknown coast was, by an odd trick of time, an English civilian of the Hon East India Company, Charles M Whish by name. Whish's paper carrying the abbreviated title "On the Hindu Quadrature of the Circle", submitted to the 'Royal Asiatic Society of Great Britain and Ireland' on 15th December 1832, did not advertise his importance as the discoverer of a strange hinterland. There was little in the title of the paper to assure its readers that the material offered to them had with difficulty drawn from that stock of mixed mathematics which the children of Kerala had till then looked upon as its exclusive property...
Pre-19th Century Transmission of Knowledge
The hypothesis that Indian mathematical knowledge was directly transmitted to Europe before the 19th century re-discovery was the premise behind Joseph's 'A Passage to Infinity'. The hyopthesis supposed that the subsequent development of mathematics in Europe in the 16th and 17th centuries would make more sense if either Jesuit missionaries in South India had acted as a conduit for that knowledge. The author investigated 'correspondence, reports, and Indian manuscripts in European archives with known or possible connections to 16th- and 17th-century Jesuit missionaries'. The conclusions the author reached are summarized by Plofker:
As Joseph candidly observes, the sifting of the various archives 'has yielded no direct evidence of the conjectured transmission'.
Joseph did leave a speculative (and difficult-to-prove) conjecture that the practical uses of Indian mathematics may have been transmitted through local craftsmen who taught their foreign counterparts. There is, however, no definitive proof of this.
Pingree's 'Hellenophilia versus the History of Science' also argues against direct transmission of knowledge before Whish (emphasis mine):
... the Indian Madhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Matthew Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Madhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the 'Transactions of the Royal Asiatic Society', in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Madhava derived the series without the calculus
Indirect transmission of knowledge is more difficult to prove/disprove. WP notes how Al-Khwarizmi was influenced by Aryabhatiya. Al-Khwarizmi and his successors developed these ideas, and built their own traditions of mathematics on top of what they borrowed and learned from the Indians. This Arabic tradition would slowly be incorporated into European scientific world-view. Yet, there did not seem to be much acknowledgement of the debt to Indians in this or mentions of their achievements.
Perhaps an example of this can be found in the numeric system with Fibonacci noting that the system of numerals came originally from the Indians, but the wider European community accepting it into common usage as 'Arabic numerals'.