In the opening scene of Saving Private Ryan, a priest wearing military gear and a helmet with a cross on it is shown giving a dying soldier his Last Rites.
Were these priests actually part of the military like medics were, or were they civilians along the lines of combat photographers?
Military chaplains were, and still are, commissioned officers of the armed forces. They wear miltary uniform but count as non-combatants much as medics do. This is the case for the American and British forces- and the Germans too. I have an idea though that the French chaplains were un-uniformed at least in WWI.
Here's the history page from the British army chaplain's department.
Yes, there were military chaplains. For instance, Chaplain Wood, of the 82nd Airborne, made all four division combat jumps and also captured some Nazi materiel during the war, which is now on display at the 82nd Airborne Museum on Ft Bragg, along with some other accouterments of his office. I'm having a tough time locating my books on division history, so I can't give you more specific information than that, but there were in fact chaplains of all faiths across the military, serving in exactly that capacity during WWII.
It's worth noting that according to the Geneva Convention, Chaplains are not allowed to participate in combat - so they are not really 'combat priests' in this sense.