This is the one point in your life that you have time to prepare for the rest of your life. How you spend your time now will set the stage for what kind of a life you will have in the future. I've noticed that the happiest teenagers are the ones who aren't dating. It introduces a lot of drama into your life and can shut doors for you down the road. Is the issue that your mum won't let you have any friends, or that she is just not giving you large amounts of time to hang out with friends?
Make connections to other people. If you are lonely, start working to cultivate the relationships in your life that you are allowed to have - immediate and extended family, maybe your family's friends etc. There are a lot of people who are usually waiting to be in your cheering section, if you include them in your life. That means responding when someone sends you a card or an email, picking up the phone when you have particularly good news or bad, spending time with them. This will give you a base to work from so that you are not choosing a girlfriend out of a place of insecurity and need when the time comes. It is also a good way of practicing your ability to contribute to other people's lives, resolving conflict, showing interest in other people, taking advice etc. A girlfriend's job should not be to prevent you from feeling lonely. That is usually what gets so many adults into the fixes they are in in the first place. You need to start with a life you are happy with before adding someone else to the mix. Don't limit yourself either to just the people around you. You can learn a lot about people from biographies that can impact and inform the way you choose to live your life. Life is too short to make all the mistakes yourself. Watch people and learn from them so that you can walk around some of the more obvious ones. This is your time to develop communication skills and to develop a decision-making process that includes seeking out wise counsel.
This is also the time of your life to gather up some skills. That may not all be formal training, but there will never be a time again in your life where you are as free to spend time learning new things. You don't know when these skills will come in handy. Many of the jobs I have been hired for have had to do with some skill I had which was not part of my main university training. This is also the time for you to see what other jobs are like. People are going to be asking you soon what you want to do with your life. Instead of just being pushed into doing something that you doesn't interest you that much, but that you are familiar with, see if you can get to know people in various fields that interest you. Get a first aid certificate, take a weekend course, get involved in something that allows you to become comfortable with public speaking, join a sport, learn to build something, learn to play a musical instrument, get involved with extra-curricular stuff at school. Find a place where you can volunteer and learn from the people you are working with. Start a file where you can keep track of certificates, dates that you have taken courses or been involved in something. This will be invaluable on your resume, and just generally in your life. Secure women want someone who has skills and interests of their own, not a man who is looking for them to make him a life.
This is also a time for you to earn and save money. That way you will have the most doors open possible to you for travel, education or getting something that you need. Educate yourself about money before you bring someone else into your life. Open an account where you can put a percentage of money that you get. This is also a great chance to develop some skills at the same time as earning money.
I am convinced that success in life comes from:
1) A foundational worldview from which should come your philosophy and decion-making.
2) Strong attachments to caring adults so that you learn to accept having your needs provided for as well as learn to provide for others' needs and so that you aren't choosing out of a place of need. (These people would preferably be your parents, but if they're not going to be it, look for someone who can fill the bill)
3) Character, confidence and communication skills (which all greatly impact each other). At this point in your life, you are likely to be insecure and figuring out who you are, simply because of the amount of time you have been on this planet. You are preparing yourself for not messing things up, when you do have the opportunity to choose your circle of friends and a girlfriend to share your life with.
That's a lot of work, none of which can be enhanced by someone to date at this point. It seems likely to me that your mom is probably concerned about some of these factors. The fact that you are paying attention to them, would probably cut some ice with her as well.
Your job is to do the next thing. If that is homework, do a good job of it. If it is helping out around the house, do more than has been requested of you with a good attitude. If it is going on a family holiday, think of how you can contribute to making the atmosphere pleasant and making it work for everyone. I think if you do these things, you will find yourself more occupied, less lonely and getting ready for a terrific decade in your 20s.