The conversation goes wrong when it sounds like a conversation about dying, or worse, about money. It goes fine when it sounds like a conversation about where things are kept.
That is not a trick. It is genuinely the useful part. You do not need to know what anybody is worth or who inherits. You need to know where the will is, which pension is with whom, and who to call.
Openings that tend to work
Each of these makes the conversation about logistics rather than about them dying, which is the difference between a conversation and an argument.
Pick the one you are facing
“I have been sorting out my own paperwork and realised nobody would know where anything of mine is. Have you done yours.”
“If you were both in hospital for two weeks, I would not know how to keep things running. Can we write the basics down.”
“A friend has just been through this for their parent and it took months, mostly because nothing was written down.”
Start with yourself
The single most effective move is doing your own first and mentioning it. It removes any suggestion that this is about their age or their health, and it gives you something concrete to show.
It also means you are asking them to join something rather than to submit to it, which is a materially different request.
Ask for locations, not contents
Where the will is, not what it says. Which bank, not the balance. Who the lawyer is, not what was discussed.
Almost everybody is comfortable sharing locations and uncomfortable sharing contents, and locations are what actually prevent the months of searching later.
Expect it to take several conversations
Trying to complete this in one sitting is how it becomes a confrontation. Getting the will's location this month and the pensions next month is a completely normal pace and considerably more likely to finish.
If somebody shuts it down, that is information rather than a refusal. Try a different entry point later, or a different person: parents will often tell a sibling something they will not tell you, for no reason either of you could explain.
Write it down at the time
The most common failure is having the conversation, feeling relieved, and recording nothing. Six months later you remember there was a lawyer and not which one.
What to capture is in the if something happens to me file.