It has to do with whether it can pay its debts or not.The NHS does not have debt in any meaningful sense, the government takes on debt, and chooses to spend some of that on new missile systems, some on wars in foreign countries, some on education, and some on the NHS.
A part of government cannot be bankrupt without the entire government itself being bankrupt, because it is funded as part of government obligations, and the amount given to it is decided almost entirely on the whim of politicians - it varies considerably and depends on all the other expenditures. You might argue that the UK as a whole is borrowing against its future rather than paying down debts (and it is currently as many countries do), but to argue that the NHS is somehow in debt or in danger of bankruptcy simply doesn't make sense when it is part of a variable budget paid for with taxes and borrowing.
You might claim the UK can't afford the NHS, but I'd point to the other parts of the budget which consume considerably more money, and compare it to the spending of other countries - that's a very different discussion, and a valid one to have, but it's not one you can have on the basis of the NHS somehow being bankrupt as compared to an arbitrary budget set by politicians with an agenda.