Relax
Our youngest (now aged 7) was diagnosed by the NHS as Autistic aged 2. Obviously we've done a massive amount of research to support him. He is extremely high functioning but we were told the NHS don't usually subdivide into high/low, etc. He essentially had Aspergers with a speech delay but now he's a complete chatterbox, and while his social skills still need work, he's basically up there with the rest of the class - mostly streets ahead especially in areas such as maths.
As you will know, Autism is a spectrum disorder and there is some evidence that there is a genetic component in some cases - but this is far from proven in the same way and to the same extent as hereditary physical conditions. There are are all sorts of contributory factors (older fathers like me are said to be more likely to have autistic kids) and as far as I know we're not at the stage yet where insurers and assessors can reliably and predictively take it into consideration based on family history to the extent of excluding people from employment based on a relative with ASD. As far as I know Autism is NOT classed as a mental illness or a psychological disorder so the premise of mentioning it in respect of a family history of mental illness is false.
So while it makes sense to obtain a copy of your GP records to know what they know (if anyone requests a copy of your medical records you should have to give your consent and specify whether you want a copy of the information supplied - that is your right in law) I would advise against fretting about it or making a big deal about it. The more you get stressed about it the more others are likely to start to get stressed about you getting stressed about it.
Just chill and you will be fine.
Good luck with your application.