The British press certainly loves their precious Queen, enough to follow her ongoing daily events up to the minute. I am certain they would analyze every scrap of information obtained by their minions of paparazzi cameras and be able to pick out the physical signs of any real serious ailment.
As the Queen ages it is getting harder to keep the British press from knowing if she is gravely ill. The press has a way of finding things out by using their many sources. Simple observations of the Queen would also give hints as to whether she was simply sick or was much more seriously ill.
The British press will only know what the Royal family chooses to tell them. If Queen Elizabeth were gravely ill and the family wanted to keep it quiet, then the press will probably never know otherwise. At best they won't find out for a very long time until someone comes forward with more information.
If the UK is anything like the US, then medical records are private and only information the patient wants to let out is let out. Also, the royals keep any secrets they personally feel they need to. The queen is one of the most powerful women in the world. She will update the world at some point soon.