How long did it take for their works to get published there, was it censored, were people persecuted for promoting their ideas?
And I guess if anyone wants to answer a more general question: In general, how long did it take for the books of the scientific revolution to get translated and published in places like the Ottoman empire, and were they accepted quickly, ignored or actively suppressed?
I'm going out on a limb to say you're not going to get a quality answer. I'm not aware of anything interesting happening in Islamic astronomy by 1400. And this is a subject I've read much more about than I really ever wanted to. The Ptolemaic system remains in Islamic astrology up to that point, I'm aware of writers mumbling 'the Qu'ran says so' (but that's no excuse), and in the early 1000s I know there were objections raised to the particulars of Ptolemy's system (although not the essential problem with it).
Muslim scientific advancement, in every category, slows about that time. The reason and mechanisms of that stagnation are a can of worms unto themselves.
Of interest to you is probably Indica by al-Biruni which mentions an idea of heliocentrism out of the east, with Indian and Persian contributions. He dismisses it basically off-hand.
al-Biruni's response is in no way out of place in responses to heliocentrism even in Europe - it was a cute idea that solved some problems but created even more. Tycho Brahe, a consummate scientist, has similar disagreements. The heliocentric system (before Kepler, and especially Newton) was utterly impractical because it failed to create useful predictions while rocking the boat of eveyones' worldview.
The Ottomans got both Kepler and Copernicus in the seventeenth century and they were, to the surprise of absolutely no one, skeptical. Their astronomical tables were still good, and so they were able to debate but practically ignore these developments. They didn't reject them, per se, but there was certainly ideological backlash against them, and so they were not as interested as the West would become after Newton.
To piggyback on this, if you don't mind, how did the Islamic world react to the findings of Isaac Newton? (Feel free to pick anything you're familiar with.)
I think the fate of the Istanbul observatory of Taqi ad-Din which was built in 1577 shows very well how these works were treated in the islamic world. It was just before Galileo and was destroyed 3 years after it was built.