Stats,
Before I dip out of this completely can I just put a few lines down to try and explain from my point of view?
You are asking in effect what would change minds about vaccination and compare it, not positively, to religion.
For me I think it would take some acceptance of the fact that vaccination issues do exist.
My mother had a massive lump on her arm and was very unwell after smallpox vac. I still have scars on my arms following vaccination and I too was very unwell. In bit instances we were effectively tufted at.
Ds1 had his vaccinations and was unwell but again I was told it was not a problem and I had huge faith in the whole vaccination programme so had ds2 vaccinated.
His health increasingly crumbled and he was really unwell after the batch at 18 months and never recovered. He has many problems now including autism.
My nieces were then vaccinated and one had a very serious seizure whilst still in the doctors surgery. Literally as she was being treated the nurse was trying to insist to my brother that this was simple coincidence.
I could add many other things from my family but I won't bore you, but it seems to me that something about my family and vaccinations doesn't mix.
I am content that we are in an unfortunate minority and also that we are a bit weird 
But what would be really helpful would be for the science/medical community to not treat me like a twat .. Or indeed some kind of guidable, foolish zealot.
I would love someone to actually look at my child and try and figure what happened and why we can't just get vaccinated without it being a problem.
Dealing with ds2 is tough and, having had dd the decision to not vaccinated her remained tough. It is much tougher when I constantly have to endure the sort of stupid twatty judgement and the threats of her being denied schooling or forced to leave the country by people who feel I should not be allowed to parent this issue according to my very vivid family experience.
It rather makes me question who the zealots are.
I would love to ve the mum who just rolls up to baby clinic and has no qualms. But I genuinely believe that anyone who would vaccinate my dd would be putting her health at great risk. Because funnily enough, having not been vaccinated she has not the healthbproblems that bothered her brothers nor does she have any asd or sn issues.
You compared the parents of unvaccinated children to a religious cult. From where I sit the medical community responding to my children with ' it didn't happen, you just imagined it, it is just co-incidence' are far more cult like, in particular in their refusal to accept and different opinion without abuse and name calling.
It is quite hard to take sometimes tbh.
I hope you can read in my tone that I am not seeking to be combative. But you seem to be trying to understand. I am just trying to explain why my choices, as a parent, have to be different from mainstream. And why the name calling. -be it cult members or fools or twits who believe any shit on the Internet - is challenging for me.