I think that I'm like you, tbh. I can distinctly remember that hideous cold, creeping feeling up my spine when I used to think about it from when I was aged about five. As for you my mom used to say, if I asked her, 'oh don't worry about that' but, of course you do. The ony real way I used to be able to not think about it was to read a book until I fell asleep (and then ended up having to find my page again in the morning!) This pretty much stayed with me until I had my own children; probably because I was so busy then, and still am now, that I don't have time to think.
It's probably a mistake to think that your mom didn't think about it - we're all human - death is the only thing that really binds us all together - but it's a difficult thing to explain to children.
When my wife's mother died and left our five year old heart broken I went for a slightly different approach: I explained how children are a product of their parents and part of her mom and dad are part of her (all scientifically true - which made life easier for non-religious me
) - and that our parents are part of us and so on, and through that part of her nan was in her; seemed to make sense for her at the time.
Ultimately - I think you just have to keep yourself busy and think about other things. Worrying about it doesn't change it (and the alternative of immortality is fairly bad too when you think it through) and is just going to screw up the years you have to live. Easy to say though and we all think about it in our own way though.