Already discussed and solved here:
https://buddypress.org/support/topic/need-activation-key/
and a bit older here
https://buddypress.org/support/topic/activation-code-after-new-member-registration/page/2/
And of my own experience, i had similar problems in the past due to using the same nav session during the registering process.
In brief, a user’s navigator cache problem.
Hope that @djpaul ask on this tread, once more time, about the server type, and that he will finally give a reason or a solution! 😉