Apple's current "Buy Mac or iPad, get a gift card up to £120" promotion says in the small print, "To verify your student or staff status, you will be redirected to the third-party website UNiDAYS". You can see details of this promotion at
https://www.apple.com/uk-edu/shop/browse/overlay/back_to_school (which normally appears as an overlay on another page, which is why it looks a little odd).
UNiDAYS discounts are generally available to anyone in sixth form, college or university. (They verify that you are via the fact that you have a school/college/uni e-mail address).
However, when you follow the link from the Apple web site says to the "Verify your student or staff status to shop the Apple Education store" page on the UNiDAYS site it says, in the small print, "Available to current and newly accepted students in higher education, parents buying for them, and teachers and staff at all levels." (Higher Education means university or equivalent.)
So it would seem that the "Buy Mac or iPad, get a gift card up to £120" promotion applies to university students only. How they establish that you're a "newly accepted" student isn't clear.
If you have a different student discount in mind, please post a link to it on the Apple site.