Original meaning: doing something in a slow, unhurried, and composed manner. Often implies a measured, gentle action.
He slowly stood up and opened the window.
祖母はおもむろに話し始めた。
My grandmother began to speak calmly and deliberately.
Modern non-standard usage meaning 'suddenly', likely arising from confusion with similar-sounding words. Many native speakers consider this incorrect; the original meaning is 'slowly'.
おもむろに雨が降り出した。（非標準的な用法）
Suddenly it started raining. (non-standard usage)
He slowly stood up and looked out the window.
She began speaking slowly and calmly.
The old man slowly reached out his hand and took the book.