"Alternately" or "alternatively"

Fraser Orr's answer corresponds to how I learned English in the UK in the 50's and 60's.

However, the Oxford English Dictionary, under 'alternate', gives its first six meanings consistent with this, but then it says

II. Senses equating to alternative adj. Chiefly N. Amer.

with examples from 1776.

And for "alternative" the first meaning given (attested from 1540) is

1. Characterized by alternation; alternating, alternate.

Which indicates that the neat distinction Fraser Orr makes simply does not hold in the real world - it does for some people (such as me); but there is ample authority for using the words either way round.

a traffic sign that reads "alternate route" is not telling you to use the opposite route as last time every time you come to the same sign on the road. this is not the same as the action 'to alternate' (verb)

an alternate route or alternate plan or alternate contingency do not flip back and forth between opposing states in any way. The only alternate plan I have is to take the first off-ramp. (adjective) Taking the first off-ramp is my only alternative. (noun)

if alternative in this sentence means 'alternate plan' then 'alternative plan' means 'alternate plan plan' american english seems full of terms from technical jargon that originated from proper english, but have come to replace it.