You can cancel your test quite late (I think its 5 working days before - but you prob want to check that). So you may as well keep it for now and see how you feel abit nearer the time. Mid-oct is still a fair way away. How many lessons/hours are you having a week? You could always see as well if you could fit in more lessons if you wanted for a bit. Maybe an extra lesson a week, or a two hour lesson rather than a one hour?
At the end of the day, it is you taking your test, so you need to feel ready for it. Not your instructor/friend/parent telling you that you should be taking it by now. I know that I got talked into taking my first test, and failed it spectacularly. That really knocked my confidence, and it took ages for me to pull it back together, and eventually pass a test.
You say you havn't done a wide corner reverse, but you think you have done everything else. Have you done...turn in the road, parallel park, bay park (depending on your test centre), emergency stop, and reverse round the corner? If you have already done some reverse round the corner it doesn't take that long to adapt that method to wide/tight corners. But again, it all depends how confident you feel with them.
Anyway, I'm rambling a bit here. Maybe you are best to do a mock test. If your instructor is part of a driving school they may be willing to get another instructor to take you for a mock test (which is often useful, to get someone elses opinon on your driving). Or if you can't do that, just get your instructor to do a mock test, it shoudl give you some idea as to how much work you have left to do before you are at test standard.
One last point, I understand you don't want to disappoint your mum, but plenty of people don't pass first time, so try not to worry about that too much. Besides, you could always start saying that the test is much harder to pass now than when she was learning (but that obv depends on when she passed!!) Maybe it would be worth not telling your mum when you have booked/taking your test. Although I told my parents when I had tests booked I didn't tell any of my friends, as they kept going on about how long it was taking me to pass my test. I didn't want the extra pressure of them asking me how it had gone.
I think thats probably enough of my waffle for now....might add extra if I think of anything!!