Borscht is also the name that I know (though as an American, I can't speak to the UK). "Borshch" is the orthographical equivalent to Russian Борщ and Dutch borsjtsj.

Borscht with a T is the Yiddish pronunciation, unlike Russian which drops the T. It's a more common spelling in areas that historically had a larger Eastern European Jewish population (like the US, where the "borscht" spelling extended out from New York).

I don't think so. I'm British and know it as "borscht"; I've never (consciously) seen "borshch" until now.

the comment was explaining which spelling would most closely resemble the original russian pronunciation. as some of the sister comments explain, borscht is the yiddish pronunciation

Borscht is the pronunciation in Bulgarian of Борщ.