https://www.citizensadvice.org.uk/work/leaving-a-job/dismiss....
I think you're misunderstanding the page. The employer is always required to provide a reason, whether you worked somewhere 1 week, 1 year, or 20 years. It's just that within the first 2 years of employment you don't have the right to challege the dismissal UNLESS it was for a discriminatory reason. But yes, no employer ever would say "we're letting you go because you're pregnant"(or maybe they would).
>> It's "at will" employment like the US.
Well, not quite. They still have to legally give you notice(only 1 week is required legally if you worked less than 2 years, but yes).
What you cannot do is challege as dismissal as unfair for any non-discriminatory reason. It has absolutely nothing to do with what the employer wrote down as the reason for dismissal, it's the question of what grounds are you challenging them on.