Ok, you can have 'High' Schools in Britain, but this is not their formal appelation.

Formally, British secondary schools are: Public (a misnomer, since they are actually private); Grant-Supported (private again, but with stipends for bright children from less well-off families), Grammar (an anachronism from the time before the 1950s, but some survive: secondaries for the children of the middle classes); and Comprehensive (for all ability levels, introduced with the 1944 Education Act). There also used to be Secondary Moderns -- dump schools for children who had failed the so-called Eleven-Plus exam.

Anyway, all this is beside the point. Do you say Висша школа in Bulgarian? No!
Therefore, you may not translate гимназия as High School, period!