First off, the timing of your results isn't unreasonable as most resit exam boards take place in early to mid September.
It sounds like you should put in an appeal but you will have to be able to provide strong evidence for what you have described. This might include documents relating to your mum's illness (with your mum's permission, of course), such as appointment letters and/or a letter from the family GP explaining the situation.
If the appeal is upheld, you will probably be given another chance to take those modules. The academic regulations (which you should consult) will dictate under what conditions, such as repeating them on their own.
I would also suggest that you consult the university regulations on the the appeal process, as well as seeking advice from your SU about how to prepare it.