Hi there,
Thats great you've started preparing with savings! There is no amount which is "enough" for university. I know people who have started with a lot of money and people who have started with nothing - myself included. Its all about budgeting and managing while you're there. I think 2K is a great start
I recommend taking out some student loans to help covering essentials. You could also talk to your univeristy about additional financial support available and maybe even look into a part time job.
I understand the hesitance in asking for money but if your mother is offering/happy to there is no shame in accepting the help!
Hope this helped,
- Sophie (uni of Bath)