Do I have to tell my insurer I am in hospital?

No, not if you're going to pay the bill with your own money. There's no law that says you HAVE TO use your insurance.

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But if you intend to use your health insurance for your hospital stay, you should inform your insurer right away. You may need pre-authorization for some things. If the hospital already took your insurance card, it's likely they started a claim for you already.