For people to demand anything you'll need a democracy. For people to want universal healthcare they need to understand that socializing the health cost is a benefit for the population.
Add the words together and you get social democracy.
In most countries this is what lead to universal healthcare. In most countries the social democracy parties were founded by labour unions.
Despite USA being first with labour unions, they never really succeeded, because they were violently struck down early on.
Anyway, it's that simple: Join a union, let the union establish a political party, let the party make universal healthcare. I know that seems very uphill, but it doesn't actually have to take centuries to do.
When the first unions were formed in Europe, the workers also expected it to be a multigenerational battle, and yet decided to try it in the vague hope that it might eventually benefit their grandchildren. However they were so successful that they achieved the goal within their own lifetime.