The related idioms for "gradually showing a warm and caring side over time" are:
- warm (up) to somebody: to start to like someone:
I wasn't sure about Sarah at first, but I warmed to her after we'd been out together a few times. Cambridge
- grow on someone: if something or someone grows on you, you start to like them more Macmillan
However, I don't think there is an exact equivalent in English to describe the character. Tvtropes offers some alternatives for tsundere though:
Punchy Love, Oranyan, Hot And Cold, Bitter Sweet Heart, Affectionate Spitfire, Mood Swinging Sweetheart, Sweet And Sour Personality, Sugar And Fire Personality
* Oranyan is sometimes used to refer to a male tsundere character - incorrectly since it means the complete opposite.
Additionally, tsundere appears as an English word in Wiktionary and it is already used as a loan word:
(chiefly Japanese media) A fictional character who fits the archetype of being cold and even hostile towards another person before gradually showing a warm and caring side over time.