I can't speak to what EK would do at BHX, but two incidents on other airlines:

In 2002 a late evening flight from SYD on Thai was delayed with a last minute mechanical problem. After a few hours, due to the airport curfew, we were offloaded and departure was rescheduled for the next morning (time TBD). We were offered the choice of an airport hotel or "making our own arrangements". It was 12:30 am at that point, I live an hour (and $150 taxi ride) from the airport and they didn't have a specific time for the next morning, so I took the hotel option.

In another case, the AZ (never again!) flight from Sydney was delayed by 24 hours. This was known in advance and they were supposed to notify me but hadn't, so I arrived at the airport to be rudely told "come back tomorrow". I was the only passenger there at the time, so they seemed to have contacted most of the other passengers before they left for the airport. (They made up excuses about not having my phone number(s) and were rude about reimbursing the extra taxi fares, and then they lost my luggage for two days, but that's for another thread. Still, never again. )