What non-Roman scripts keep foreign words in Roman?

In the last few decades, written Greek uses Roman script for foreign names by default, unless the name is extremely newsworthy. So you’ll see

Το συγκινητικό ντοκιμαντέρ για τη ζωή της Amy Winehouse (The moving documentary on Amy Winehouse’s life)

Rehab της Amy Winehouse, σε διασκευή των Vocal Adrenaline. (Rehab by Amy Winehouse, arranged by Vocal Adrenaline)

More rarely, you’ll get Roman + Greek:

Ο πατέρας της, Μίτσελ Γουάινχάους (Mitchell Winehouse) ήταν ταξιτζής και τζαζίστας, η μητέρα της Τζάνις (Janis Winehouse) ήταν φαρμακοποιός (Her father Mitchell Winehouse was a cab driver and jazzman, her mother Janis was a pharmacist)

And rarely (I find), you’ll get just Greek:

Η Έιμι Γουάινχαουζ «ζωντανεύει» δια χειρος Ασίφ Καπάντια  (Amy Winehouse comes alive through the work of Asif Kapadia)

This doesn’t get done for world leaders who show up in newspaper headlines constantly. So Schäuble, the German finance minister and current bête noire of  Greece, is always transliterated as Σόιμπλε. The headline in the following is more unusual than the lead;

Κρεμλίνο: Ο Putin δεν θα συναντηθεί με τον Εrdogan (Kremlin: Putin will not meet Erdoğan)

Ο Πρόεδρος της Ρωσίας Βλαντίμιρ Πούτιν δεν σχεδιάζει να συναντήσει τον τούρκο ομόλογό του Ρετζέπ Ταγίπ Ερντογάν … (The president of Russia Vladimir Putin does not plan to meet with his Turkish counterpart Recep Tayyip Erdogan)

The foreign script is always Roman; you won’t see  Ο Путин δεν θα συναντηθεί (even though Cyrillic should be more familiar).

Of course, there is individual variation. In this Reddit thread (Τι κάνει ο Χόκινγκ; Επιστήμη ή μάρκετινγκ; (Το άρθρο που “καίει” αυτή τη στιγμή το twitter #protagon_science) • /r/greece), the first comment has Χόκιγκ, the second has Hawking. 

The untransliterated Roman is a recent thing, maybe two decades old. Before then, transliteration into Greek was universal.