Right now I am working on a piece about Romani for my Substack. The first clear mention of Romani in Europe is in Crete in 1322. This is certainly postdated their arrival, but there is a good amount of circumstantial linguistic and genetic evidence that they left the Indian subcontinent not much earlier than 1000 AD.
Genetically on the whole the Romani are ~30% South Asian in ancestry (some groups and individuals are closer to 50%, others to 20%). Even besides this fact, just physical inspection of some of the Romani makes it clear what part of their origins are. Their language is similar enough to other Indo-Aryan languages that a Hungarian Reformed theologian who learned some Sinhalese at a seminary in the Netherlands in the 18th century was the first to identify that it was Indo-Aryan because of its similarity. Some lascars did the same in the United Kingdom when they overheard Romani talking to each other in Wales.
But this brings me to a curious fact: the Romani seem to have forgotten their Indian origins by the time they arrived in Europe, after a sojourn of centuries in the Middle East. The Romani are nomadic people who have an oral culture. There is a great deal they remember. The term for a Christian “cross” in Romani is trushul. But they forgot where they came from.