How can I write a nightmare scene without making readers aware that it is indeed a nightmare and not something that is really happening to the character?
Relating Dreams and Nightmares as real life scenes in your stories are an over used trope. We can thank all those writers who came before us to stripping the trees of the low hanging fruit to tell their stories.
If the trope is important to your story, then by ensuring that your readers know its is a dream you can refute any complaints you are using an over wrote trope.
If you want you narrative to be free of distancing statements like last night I dreamt … you can have your character relate that they always dream of something specific — unicorns or candy corn or rubber washing gloves. They should react to that observation in some way that reveals plot or character. Then you can have those specific elements show up in your dream scenes and they will clue your readers in to that fact that they are in a dream or nightmare.
I've watched and read a lot of messed up stuff. Chris isn't wrong, in that you can certainly write very realistic dreams. But is that really what you want to do? I think of The Illustrated Man, Inception, or Brazil, and the surreal qualities of the settings in these works are such that the lines between dream and reality blur. Eventually, you're not even sure if it matters.
I have a character who dreams about memories, but sometimes it's not HER memories (she has a ghost). Other times, they are dreams, but premonitions. Other times, they are symbolic and sent as messages. Unfortunately, you usually know they are SOME kind of dream, but not how 'real' they are. The character also hallucinates sometimes (and sometimes the hallucinations reveal a deeper reality, making them more real than reality), so it gets murky. But short of having reality be a slippery slope for the character, readers don't want to be tricked into thinking dreams are real. The best I can suggest is to make the character's reality messed up enough that the character isn't sure what's real.
Altered mental states (I'm leery of 'mental illness' as a tag) allow your character to inhabit a very strange reality. You can portray that reality to your reader. Sometimes, their dreams might be more real than their reality (depending on what genre you're writing in).
Virtual reality is another popular approach to this. Once in a virtual reality, you can never be 100% sure your character has left the virtual reality. There's always the nagging possibility that something (or everything) is just a construct of the virtual reality.