Sharing with research team about being affected by a natural disaster

I would call it not only professional but close to mandatory. A supervisor has to be informed of special circumstances or reasons for specific behaviour in order to manage the team effectively, set realistic goals, assure the people are comfortable and to show basic human decency. You do not have to share something that makes you feel uncomfortable, and do not have to provide a lot of detail on personal circumstances. I think it is best to do so at an early stage, saying there is issue X and you would appreciate some consideration, rather than have to explain possible poor performance later after frictions have appeared.

Suffice to say that many academics have friends and family outside their country of work and are personally aware of what it means to be away from them in a time of need. I cannot imagine many supervisors who would not be sympathetic to such a discussion, or would not appreciate the honesty. In practice this may not mean more than a pat on the shoulder and some more relaxed time, but even that can be important. If the conditions allow, a leave or temporary interruption of studies can also be warranted.

If these are people you must work closely with then it is wise to let them know. But everyone will react differently. Most, but not all, people recognize the impact of such things on productivity. Most, but not all, will give you some "room" to recover.

But it is better that you inform them, to avoid their making assumptions that might be incorrect.

Perhaps you can think about informing different people differently, depending on your assessment of them and your relationship. For some, an email is enough. But for your advisor, a meeting might be better if it can be arranged.

And don't worry if they don't react. Some people just have an impoverished emotional life and don't connect to others. Some people just have poor people skills. That can even be for medical reasons.