Hey, I pay money IN to the NHS, can I have a subsidised lease car?
Subsidised lease cars are offered to essential users - in the same way a company car would be offered to somebody working for a private company doing field work.
Additionally, some trusts run a salary sacrifice car scheme but these are not subsidised by the NHS - the benefit of these is that the money is taken pre-tax and you pay tax on the cars emissions instead of the income tax, so if you take a low emissions car it can be tax efficient.
Don't worry, your tax money isn't letting staff swan around in cheap cars for the lulz.