find out my mum has gene defect called lynch syndrome which makes her more susceptible to certain cancers (e.g. 80% more likely to get bowel cancer by age 70 compared to 2% in general population)
There is 50/50 chance that this gene gets passed to me because my dad doesn't have it.
The genetic doctor won't give me blood test and say's I am not emotionally ready for the potential outcome because I am 21
How do i convince him to give me blood test on my first consultation?
You probably don't. Find another doctor. Also find someone called a genetic counselor. That's the specialty trained in handling people in your situation, and they can direct you toward resources if you insist on getting this information.