What's the word for the taste of gooseberries, unripe bananas, or betel nuts?

From what I gather "astringent" is often used for this sixth taste. That may seem inappropriate to you. In terms of taste, which can be very subjective, the word "acidic" might be an appropriate English word to convey the taste. Other English taste words might be to give quality of "brightness" or "freshness".

http://www.nithyasnalabagam.com/2013/06/six-tastes-of-food-arusuvai-unnavu.html

There no English word to describe thuvarpu in Tamil. However "pulippu" and "Thuvarpu" are entirely different taste English use sour for both.

The science is a bit incomplete on this topic, as reflected in the language. There are four non-controversial tastes in the western nomenclature: sweet, sour, salty, and bitter, and a recent addition of umami. Umami was only proposed in early 1900s, and wasn't generally accepted in the scientific community until the 1980s or so. MSG is to umami as salt is to salty.

There is a movement now for a sixth taste, as reflected in Wikipedia:

In Asian countries within the sphere of mainly Chinese and Indian cultural influence, pungency (piquancy or hotness) had traditionally been considered a sixth basic taste.

This may be the best translation.

You'll also find references to a seventh taste there, related to fatty foods.

The science is very immature, especially when compared to vision and audition. We're likely decades away from real consensus, and a stable vocabulary.

Wikipedia

In Tamil pulippu and thuvarpu are totally different tastes. In English we use sour for pulippu and astringent for thuvarpu.