Unfortunately, the NHS can't be free of politics, because it uses such a big share of national wealth. One of the main roles of politics is to allocate scarce (in the economic sense) resources.
Put aside arguments about health tourism and unnecessary procedures for the moment. I'm not denying them, but they actually only cost a fraction of the overall budget.
Most money is spent on the elderly and, to a lesser extent, premature and sickly children. If people want to save serious money on healthcare, they need to face up to the fact that we might not be able to afford expensive cancer drugs or care for very premature children - or, at least, that they will only be available for those who can afford them. Mental health services have borne the brunt of many cuts.
I don't know the answer. As a country, we need to have an honest discussion. Do we want a first class health service or not? If we do, we need to pay for it.
The same is true of schools, which are facing serious cuts, to the extent that they cannot possibly maintain what they've been doing.
Interestingly, it appears that the biggest swing from Conservative to Labour in the last election was amongst 25-44 year olds (not the students wanting their "sweeties"). These are the people, who can see what's happening in their children's schools, the inadequacies in maternity services, are struggling to get on the housing ladder, have had their pay frozen year after year if they are public servants, etc.