Getting run ragged until tempers are frayed and everyone is snapping at each other is caused precisely by the conditions they're striking against. And while these nurses may be prepared to strike, they may not have the class consciousness that would lead them to show solidarity with their fellow workers and help them out in ways like this.
The ones at fault are still the government, and frankly conditions are only going to get worse and worse until the NHS has been fully dismantled by privatization, regardless of how well the nurses and support workers treat each other. A fully functioning health system is a thing of the past, irretrievably, until after a successful socialist revolution.
You can in fact blame the government, at least a little, because the government are the controllers of the state and the state is a machine used to wage class war - a vital component of which is stifling class consciousness. And the English proletariat has been one of the most enslaved and unconscious working classes in the world, since at least Marx and Engels' time. You're right that it's down to us to talk to others, because we as communists understand the world better and so understand why we should have solidarity with other workers. The class-unconscious people out there don't have the benefit of this understanding, so they may treat their fellow workers badly - but if we write them off as irredeemable because they're not already perfect, then how could we ever hope to succeed? What would be the point of trying to reach them?