Would it be wise for me to delay this as much as possible, so I can
avoid loosing my status and job for now?
Nobody is irreplaceable. Not you, not anyone.
Unless you are self-employed, your company will find a way to go on when you aren't around. It might be easier to sit back and avoid the work of cross-training now, but eventually someone else will be doing your job. Trying to intentionally delay that capability doesn't really benefit you. If you get on top of that now, you put yourself in a position to move on to better, and more important work, rather than being burdened by always having to do exactly what you are doing now.
Your feelings are understandable. It's nice to be heavily relied upon. It's nice to be "The Guy" or "The Gal" that others depend on. But in the long run, it's even better to be "The Gal/Guy That Makes Everyone Better".
Be the one who knows the most but is happy and willing to share. Be the one who is happy to help others get better, and thus helps the team be better overall.
As a manager, I worry when critical tasks depend on one person. In business, a single point of failure is bad practice. I work hard to avoid those situations, and to reduce the risk when they do. It sounds as if your managers feel the same. Ignoring their requests to start cross-training isn't wise. If you don't do it, eventually they'll have to find someone who will - perhaps at your expense.
Once you have others trained up, you will likely find that it can free up some of your time to take on other, even more important tasks. In the long run, you will benefit from this, as will your team. You can put yourself in a position for a promotion, or for taking on more important tasks within your team.
At a minimum, you'll be in a position to take vacations and holidays on terms that meet your and your family's personal needs, rather than worrying that you can never be away from the office for a few hours. That's a good thing for your long-term health.