There are a few things that stick out from your post and this thread.
1. I know you said £62 a week was in error, but just checking here, we're definitely talking about being over the minimum wage?
2. A union, anyone can join a union - it doesn't matter if a union is recognised (well it does, but it doesn't and we'll not get into that). In actual fact the contrary to what you would expect is true, in workplaces where there is no union people who join sometimes get a rolls royce service in that they get a union official (or even above) representing them.
3. Have you been given a handbook or access to the intranet at work to find out what company policy on this stuff is? It's quite possibly company policy exists and says something such as in the event of the clocking in machine failing or being repaired, bonus will be paid where there is paper evidence, the benefit of the doubt will be given and there is line manager's discretion etc.
4. ACAS offer great advice, I doubt they will be too helpful here. They tend to shine in other cases.
5. You've already spoken to them by the sounds of it several times, you need to write a letter. Just a letter, nothing threatening or confrontational about it.
6. It hasn't been resolved through several attempts so if you want to go beyond this at this point you should consider writing a letter which puts in a grievance.
7. Complaints about HR are often dealt with by complaints to local area/region whether that is applicable here or not is not clear.
8. There is normally a reason for payroll problems, if they are happening they are likely to continue unless someone does something properly about it.
9. Stating their decision is final and there is no appeal is nonsense. Even if you made this a grievance and they repeated the whole lot you've already said you'd still have an appeal.
10. Evidence of clocking in and out - you should keep your own records. Company says we don't have any evidence of you clocking in and out and what time. They have no evidence, you have evidence, whose opinion based on their own respective evidence is likely to take precedence? Yours.