Well, the news isn't always accurate. Agenda driven news outlets have a good time creating issues out of nothing, and since they usually stand unchallenged, they can continue to report things that are out of date, inaccurate, or misconstrued.
No one knows the number of illegals in the UK. Even if we did know, we don't know the numbers that are using the NHS illegally. And even if we did know that, unless you're talking about biometric IDs, this would just mean that forgers would have another relatively easy stream of income.
If it did turn out to be a big deal, I could see putting someone's picture on their records at their GP. It might help reduce medical errors as well. IDs wouldn't be needed, and it could be done relatively cheaply digitally. If someone came in without a picture on their file, they would be treated, but would need to have theirs taken then.
But blaming NHS funding on illegal immigrants is sort of like blaming the housing shortage (both for things like council rentals and general renting and buying) on illegals. It may (or may not) be a part of the cause, but there's a lot more to it than that.