In both British and American English, 'presently' can mean:

(1) In a short time: a bus will arrive presently; the sky grew dark and presently it started to rain.

(2) At the present time: we are presently watching TV; my mother is presently in prison for drunkenness.

Presently (etymonline)
Presently (Oxford)
Presently (Oxford Learners)

Meaning (1) is the majority usage in British English, although meaning (2) has been used since the 14th century, and meaning (2) is the majority usage in American English

In both British and American English, 'presently' can mean:

(1) In a short time: a bus will arrive presently; the sky grew dark and presently it started to rain.

(2) At the present time: we are presently watching TV; my mother is presently in prison for drunkenness.

Presently (etymonline)
Presently (Oxford)
Presently (Oxford Learners)

Meaning (1) is the majority usage in British English, although meaning (2) has been used since the 14th century, and meaning (2) is the majority usage in American English. Consider also 'momentarily' which means 'for a very brief time' in BrE (a struck match flares momentarily) and 'in a very brief time' in AmE (my husband will be here to pick me up momentarily).