Look - as I've demonstrated above, the problem is due to management. If what you're saying true regarding immigrants, then none of Robinson's measures, for instance, would have worked (because even the best management couldn't cope with what your describing - but did in practice). I'm not denying that an extra influx does place extra strein on a health service, but it's hardly an enormous difference. If you were theoretically to remove all immigrant use of the health service - the strein on it would in practice be only ever so slightly reduced (and would solve none of the problems with it). And when you take the figures at a national level (i.e. bear in mind what that'll mean for individual hospitals on the ground), it doesn't amount to much.
With the right kind of changes (as Robinson demonstrated quite clearly), the NHS would be able to cope with demand quite effectively, irrespective of extra numbers.