loujo, it is hard with a baby and everyone no matter how young or old they are needs a break.
I'm lucky as I have a dh who is there to sit with the dc's if I need to do something or want to go out. It would be so hard if I didn't have that - I know, as I was on my own (through dh working away) for quite a while and I definitely needed a break.
It would be nice if your mum offered to help you rather than you having to ask her.
I know that having to ask sort of feels like she would be doing something that she doesn't really want to do, but you maybe surprised if you talk to her and let her know how hard you find it sometimes.
My MIL is totally rubbish and she's the only support (other than dh) I have here where I live, and I know only too well that feeling that you would like someone to offer because they want to see their GC without having to ask them to do it or that they can see that you're exhausted and are doing it for you too.
You are only young and if you're on your own it becomes even more important that you have a support network, but ultimately you can't force people to be the way you would like them to be so try to find other avenues of support either through your dd's other grandparents or through friends that you will need to make.
Your dd is getting to an age now where you can go to local toddler groups hopefully you will meet some good friends there that could help you out, or speak to your health visitor about the available support or groups for your age in your area.
No, you're not being unreasonable to want your mum to be more involved, and it is sad that she isn't and it's making you feel like this but things can and do change for the better in time and I'm sure that one way or another it will for you too.