Though it is too early to say, I think democratic governance systems evolve by responding(or failing to respond) to the drivers of change, and the impact on the status quo vis-a-vis the health service system depends on the magnitude of this driver (epidemic). If it is controlled relatively early and easily, I think it will result in more of a pat on the back and a validation of existing systems. But if it uncontrollable, it will result in a resetting of a lot of existing notions.There is a theory that an event of this magnitude results in a more equitable health outcomes in its aftermath, like life expectancy, mainly due to state controlled redistribution of resources (though this theory is debated, Amartya Sen has written on this if you are interested). Possible changes in healthcare in India that we may see, can be divided into two outlooks, an optimistic and a pessimistic one (according to me).

Optimistic

1. A renewed focus on strengthening government health infrastructure and increase in funding of epidemic surveillance systems and research capacities.
2. An increasing consensus amongst the politically vocal class(aka the urban elites) on the importance of keeping private healthcare profiteering in check and a demand for publicly provisioned healthcare (because of the humongous costs incurred by them during the epidemic). This may also lead to increasing demand for universal health care through tax funded/social health insurance mechanisms, esp in USA.
3. A radical expansion of the welfare state for the poor, including a more fair and equitable food distribution program, incorporation of labour rights of informal sector and some ideas like universal basic income may get a boost, at least.
4. Nationalization and decreased dependence on global supply chains for critical industries of healthcare, including pharmaceuticals, medical devices, Personal Protective equipments etc.

Ambiguous

A move away from globalism, as many local political movements advocate this, case in point being UK.

Pessimistic

1. The coronavirus epidemic will be used as a case to advocate the "efficiency" of private sector in coming up with vaccines and other solutions, thereby gaining more leverage against governments.
2. State may further shrink away from health provisioning, and the private sector will take the forefront, with the former acting as a regulator.

Actually I got bored now, but there are more changes to be expected. Will follow up later.