Acting on the sensible suggestion proposed by Edwin Ashworth some 30 months ago (in a comment beneath the posted question), I consulted several British dictionaries to see how they define alternate as an adjective. Here are the results.
From The Penguin English Dictionary, revised edition (1969):
alternate adj (of two things) following continuously one after the other; arranged alternately; (bot[any]) growing along an axis, first on one side, then on the other
From The Concise Oxford Dictionary, sixth edition (1976):
alternate ... a. (Of things of two things of two kinds) coming each after one of the other kind (alternative leaves, angles, those placed alternatively on the two sides of stem, line) ; alternative
From Longman Dictionary of Contemporary English, new edition (1987):
alternate adj 1 (of two things) happening by turns; first one and then the other: a week of alternate rain and sunshine 2 one of every two; every second: He works on alternate days. 3 {[used only before the noun it describes]} esp[ecially] Am[erican] E[nglish] instead of another; alternative: an alternate plan/suggestion
From The Concise Oxford English Dictionary, revised tenth edition 2002):
alternate ... adj. 1 every other (of two things) each following and succeeded by the other in a regular pattern. > Botany (of leaves or shoots) placed alternately on the two sides of the stem. 2 chiefly N[orth] Amer[ican] another term for alternative.
And from Collins [online] Dictionary (current):
alternate ... adjective ... [1] Alternate actions, events, or processes regularly occur after each other. | They were streaked with alternate bands of color. [2] If something happens on alternate days, it happens on one day, then happens on every second day after that. In the same way, something can happen in alternate weeks, years, or other periods of time. | Lesley had agreed to Jim going skiing in alternate years. [3] You use alternate to describe a plan, idea, or system which is different from the one already in operation and can be used instead of it. | His group was forced to turn back and take an alternate route.
The context in which these British dictionaries acknowledge alternate in the sense of "alternative" ranges from not recognizing that the meaning exists at all (Penguin 1969) to characterizing the meaning as a chiefly North American phenomenon (Longmans 1987 and Oxford 2002) to not attaching any region-specific label to the definition (Oxford 1976 and Collins current).
The upshot of the definitions seems to be that whether use of alternate to mean "alternative" is mostly North American or universal in English, it is not exclusively North American—and this in turn suggests that there is at least some level of use of alternate in this sense in the UK.