Transiting from Dubai via Madrid to a local flight in Spain operated by Iberia. (Terminal 4S arrival, Terminal 4). All tickets issued by Emirates. Original transit time 2.5 hours, now down to 25 minutes. Iberia have advised me of the the flight change as a FYI (but have not mentioned the transit time might be an issue etc). I've heard nothing from Emirates although I can now see my itinerary has changes. Is it always upon the traveler to work out that there might be an issue (25 minutes is to short to transfer from 4S to 4 in Madrid) after changes in connecting flights (even if they are on a single ticket)?
For a schedule change in advance it is always the ticketing carrier (in this case, Emirates) that is responsible for rebooking you after the change.
In a perfect world this would be handled automatically, but when multiple airlines are involved it generally doesn't - especially when those airlines are not part of the same alliance (Emirates is NOT OneWorld, despite your questions heading). I would suggest reaching out to whoever you purchased your ticket from (either Emirates or the travel agent) and asking them to arrange an alternative. It is possible they will contact you, but the earlier you arrange the alternative the better the options may be.
Depending on how much notice you were given of this change and how much you need to change your arrival time, you may be able to claim compensation under EU261.