Assuming you paid ‘Air Italy’ just once, the good news is that your contract with ‘Air Italy’ is to get you from Delhi to Rome and back again. If they cancelled the DEL-MXP legs, they should be offering you an alternative. They have NO responsibility for the Ryanair flights or the Indigo flight to Hyderabad, assuming you paid separately for these. If ‘Air Italy’ can reschedule you to arrive in Rome before the Ryanair flight to Athens (and also reschedule your departure from Rome after your Ryanair flight arrives), you might be able to keep the other flights.
To echo some of the other replies and comments: Why would you want to fly such a complex routing, when many airlines seem to offer Delhi to Athens with just a single stop? A complex routing like this is big a chance for something to go wrong: delays, cancellations, lost luggage, endless hanging around in foreign airports, and perhaps the need for a transit visa as well. An added complication is that once in Rome, you would need to collect your luggage, transfer to another airport and check in your luggage again. Also, if you are a non-EU passport holder, flying Ryanair has the added difficulty of needing to visit their special desk so your visa can be ‘validated’ before they will let you board.
Your only option is to pick an earlier flight so your itinerary is not messed up.
In Europe we have usually a very good compensation system for denied boarding and cancelled flights. You can check up on the rules if you search for passenger rights.
SADLY, it does not apply to you, since you have booked each flight on their own and also the airline informed you well ahead of time (> 14 days)
The General rules for refunds in Europe are as follows (Any flight leaving Europe and any European Airline flying to Europe):
Cancelation < 14 days of flight / denied boarding (even with alternate flights, but refunds may be reduced if you get there in time) / Flight Delayed >3 H :
Claiming these benefits you need to contact the executing airline. I suggest you do not use any 3rd party website in order to claim, since they take a cut of the refunds and its only 2 emails or so to get the compensation.
Unforseen events may void this. Plane breakdowns are not unforseens/unexpected since they happen all the time.
I’m afraid that, because you booked your Air Italy and Ryanair flights separately, your connections are not the airline’s concern.
Air Italy is obliged to offer you a choice between alternative transport to your destination (which does not have to be on the same days etc) or a full refund. That’s it. You can try to negotiate for better flights than the ones they’re offering you, maybe even an alternative connection straight to Athens, but at the end of the day you can’t force them. You are not entitled to any compensation.
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