Hi Sarah
yes your mum was very brave, and still thinking of her lovely daughter's , in many ways your mum got her wish she did not want to spoil your Christmas, no one can prepare you for your mums death, but from what you say there was an acceptance on your mums part.
She might not be sitting at the Christmas table, but she will still be with you,
I would not think about what amount of time , people said your mum had or how quickly it developed, I have a believe that people choose there own time.
My condolences to you and your sister, I hope you find some comfort in your mums last holiday, plus she chose her own time, and did not want her lovely daughter's to be unhappy at Christmas.Love Eve