This topic is not exactly my expertise, but I will try my best.
In surah 18:9-26 of the Qur’an, a story about a group of youths is told, they are known as “the companions of the cave”. The companions of the cave were being persecuted, so they fled to a nearby cave. A miracle takes place in which they fell asleep and stayed sleeping for three hundred years. When they woke up, they had no idea they had been sleeping for so long until they went back to their city and noticed how different it was. They thought they had only been sleeping for a day, but they were in for quite a surprise when they woke up and found their city in such a different state than what they were accustomed to.
Allah appeals to this story as a sign of the Day of Judgment (see surah 18:21). In other words, when we wake up on Judgment Day, it won’t feel as though years had passed-- it will feel like their had been only a day or part of a day since our respective deaths.
What happens after death is quite simple. We muslims believe that a dying person, at the moment of death, will be held in a state that’s neither in Hell or Paradise; it’s known as “the grave”. In the grave, you won’t have a physical body, but you will still be able to think and talk. When someone dies, two angels will come to him/her. They will ask three questions: “Who is your Lord?”, “what was your religion/lifestyle?” and “Who was your prophet?”.
A person will only be able to say what is truly within their heart, so you won’t be able to lie. If a person passes the test and answers “Allah”, “Islam” and “Muhammad”, they will get to see a glimpse of Paradise and will get to smell its beautiful fragrances. They’ll have to wait for Judgment Day, though. Their souls will go back to sleep until Judgment Day. If a person doesn’t answer those three questions perfectly, they’ll be shown what Hell looks like, but will have to wait until Judgment Day to finally be sentenced to its punishments.
As I said before, when your soul is asleep, it won’t feel as though years have passed. Judgment Day may not be until millions of years from now, so you won’t have to be conscious and wait until then. When you wake up on Judgment Day, it will feel as though you had died only moments ago.
I don’t believe it’s fair to compare this doctrine to annihilationism as is taught in SDA or JW teachings. They believe that a person is annihilated forever and will not face conscious torment. We believe Hell consists of conscious torment; some of which is physical and some is mental. According to surah 4:56, some people [in Hell] will have the entirety of their skin burnt, recreated and then burnt again. This will go on for eternity. There are tons of graphic images in the Qur’an about the nature of Hell.