I have a situation where a promotion came up in our company, it was 1
that I or others could have went for.  I am not saying I would have
been the best candidate or that I would have expected to get the job.

But the question is this:

If I had wanted to apply I could not have as it was simply given to
this person without advertising or discussing with other staff. 
Potentially that was a promotion that would have taken my salary from
£35,000 to £45,000.

Can someone find me examples of this situation where others have been
in the same situation and taken action against the company and what
the result was.  I am
really looking to see if it would be worth making a fuss about this.

Please only post cases that happened in the UK..

What makes you think the position has to be opened up?  Obviously
management knows your work product already since you and the one
promoted both worked there.

Smart organizations and companies do not waste time and money on
'beauty contests' when they have already found the candidate.

Count yourself lucky that you are working for a smart organization.

Really, it depends WHY you weren't offered the job. Basically
speaking, an employer can promote who they like, and they are under no
obligation to advertise the post internally or externally. BUT if you
think you weren't considered because of discrimination of some sort,
i.e. because of your sex, marital status, race, religion or belief,
disability, sexual orientation or status as a 'trans' person, then you
have a claim.

Under some circumstances employees have been able to show that they
had such a legitimate and concrete expectation that they would be
promoted that failure on the part of the employer to do this actually
constitutes an actionable breach of contract. This is a high bar to
cross however, and rare, and what you've described probably doesn't go
far enough. Sorry.