I have interviewed for delivery and operations roles for years, and I have hired them for three companies. The CV does one job: getting you to the interview. Everything else you learn in the room.
The question that separates everything
Ask: "describe a project that went wrong. What did you do?"
Not "what happened". What did you do.
Someone who has delivered starts with something concrete: the client refused the date, or accepted it because they had no alternative. They describe the moment, and then what they did in the twenty-four hours after it.
Someone who has chaired meetings starts with a theory. "We did a root cause analysis." "We set up a crisis committee." Both sentences are compatible with having not been watching at the time.
The three follow-ups
After the answer, three questions. If the interview ends there, you already have your information.
"Who made the final call, and when?" If the answer is "the team" or "we decided together", ask who sent the email with the date. A delivery manager who does not know who decided is not managing deliveries.
"What did you trade away?" The question almost nobody asks. Every time you saved a date you gave something up: margin, scope, another client. If the answer is "nothing", they have never managed a hard delivery, because it is not possible.
"What number did you look at every week?" If the answer is "the burndown" without you having to specify it, they are good. If they have to search for the word, they do not have their own measurement system.
The signals worth more than the CV
They can say no with a reason. In an interview, "I can do it, it's a commitment" is a weak answer. "I can do it if you drop Monday" is a strong one. The best delivery manager I have worked with said no to half of what was asked, and that is the only reason I trusted him.
They have a number that embarrasses them. The best professional can tell you their worst result and why it happened. If the interview ends only with wins, either the work was easy or they were not there.
They talk about their mistakes before you find them. If they say "let me tell you about the mistake that cost me a client", you believe them. If they only admit it when you ask, they have already forgotten it.
The final question, the one nobody asks
Close with this: "on day one, what do you look at?"
Answers that impressed me: "the previous project's documents, to see whether they were current". "the decision log, if one exists". "I ask the three people who hold the project memory whether it is still valid."
Answers that killed it: "the backlog". "the team, to see how they work". "the environment, so I'm operational".
The first person arrives looking for the why. The second arrives looking for the what. The why is what saves you the first month.
A note on the Italian market
On this market I see a problem I did not see in London: people are almost always hired on cost, and the best-looking CV wins among equals. The result is that you pay the wrong person for two years before you notice.
One approach that has worked for me: two interviews, with two different people, and the same question to both. Answers that do not line up are the most informative signal I have ever collected. It does not necessarily mean there is a problem: it means the company does not know what this person does, and you would find that out eventually anyway.