my gf was offered something along the lines of £60 a week whilst training in some textiles thing but this was all legit etc, it was only that low because for certain things i dont think there is actually a minimum, its whatever the employer can afford, which wasnt much in this case, she found something better where they offered her alot more though, but as with alot of training positions she got shafted, she got 2 weekends a month off. Needless to say that didnt last all that long.
I think getting (at least) full minimum wage for your logged hours, and embarrassing the company in court should be your main objectives.
If / when you win, it will be up to the judge to rule on any additional hours. He may find in your favour on that also, but it will depend on the evidence you have. Will the manager(s) that witnessed your extra hours testify against the company? If not, it might be worth forgetting them.
What actually happened is irrelevant, from a legal perspective. What you can prove is the only thing of importance.