Was this an internal appeal ?
How soon did you start working for the other restaurant after the dismissal?
By accepting your appeal they have admitted that they were wrong and in breach of your contract.
Accordingly they owe you the money you lost over that one month and a half as a result of their breach of contract and you can either sue them in court for it or take them to an Employment tribunal for Unlawful Deduction of Wages.
Threaten them with both
Yes you write to them giving them ,say, 14 days to pay or you will go to the court/tribunal.
The HR dept should recognise the threat.
You can write it, remember to keep a copy