I think that it is doable, if the restriction doesn't require the dessert to be exactly a tiramisu.
Many dairy-based cremes are interchangeable, similar in texture, and require no eggs at all. The alcohol in some tiramisu versions can be safely left out. The short time is the worst restriction. Thickeners like agar agar may not set in the short time.
You should start with buying prepared lady fingers, as you can't bake them so quickly, and it is probably too much effort anyway. You can use any tiramisu recipe for the moistening liquid; if it includes alcohol, just leave it out. You can add a few drops of alcohol-free rum essence if it called for rum or another hard alcohol, or a fruit aroma if it called for a fruit liqueur. If it only specified coffee liqueur, leave out the liqueur but use normal coffee. You might want to use instant coffee dissolved in water instead of just-brewed espresso, because hot coffee will interfere with the short chilling time. Then you need the dairy creme, and you can finish with the traditional cocoa powder.
For the dairy creme, you have to choose one which will hold its shape without thickeners. You can add agar agar or carrageenan or the calcium-sensitive type of pectin, but don't trust them to turn a runny mass into a firm one, the time is too short. Use them as an insurance, and hoping that you will get something which gives some resistance to being cut or chewed, as opposed to being just scoopable. Whipped cream together with Greek yogurt is not a bad idea; other cultured milk products can work too, if they are on the firmer side. I have had similar desserts made with just schmand as the filling, but they stayed overnight so some whey could diffuse into the non-syruped cookies, making the cookies softer and the schmand drier. If you have access to the type of modified starch which will thicken without having to cook (for example sold as "cream stiffener", but also as a cream replacement product to be whipped with milk), using some of it may help. Don't use liquid sweeteners like corn syrup or honey, stay with sugar, and separate the whey from the yogurt to get a firmer result.
Another way would be to use a starch thickened pudding instead of a custard. This is a more common substitute, but normal starch has to be cooked to a boil, and will require longer chilling. You may be able to pull it off, if you make the whole dessert rather flat and wide, use a pre-chilled vessel and dip the ladyfingers into cold liquid, and then put the poured dessert into a freezer for a quick chilling. It will need quick work (so lots of time remains for chilling) and frequent attention so it can chill without freezing. The first version is probably less stress.
Whichever you choose, such a major recipe change is not easy. Make a prototype at home before you cook it for a crowd, so you can catch the major problems in an early version and relaxed environment, and then you will know if it is doable for sure.
Candymaking is extremely sensitive to temperature. If the mixture heats higher than the point that produces the desired texture, you're basically out of luck. That makes it very critical that you reach and not exceed your target temperature.
This is particularly difficult when making candy because much of it starts as a sugar-and-water syrup. Water has an especially high heat capacity, meaning that it takes a relatively high amount of applied heat to raise its temperature. This is why a big pot of water takes a long time to boil. It also implies (because diffusivity is calculated from heat capacity and density) that it takes a while for applied heat to diffuse through the entire pot; that is, the bottom of the pot where heat is being applied will be significantly warmer than the top, absent more active circulation like stirring or the pumping action of an immersion circulator.
Anyway, "boiling" implies greater heat application than "simmering". The more heat you're applying to the bottom of your pot, the greater the difference between the top and bottom. If you're then measuring the temperature at or near the top, you may be getting a wildly inaccurate measurement of the "total" heat of your syrup. Once the heat diffuses completely, the mixture will be several degrees warmer than you measured, and that can completely ruin an entire batch of otherwise-delicious fudge.
Simmering is recommended because it gives the heat being applied to your syrup more time to diffuse, which means more even and consistent heating, which means you can reliably hit your target temperature without going over. If you became some kind of ninth-dan fudge guru, maybe you could boil and know exactly how much heat is enough to hit your target, but don't count on getting there.
The sensitivity of this process is part of the reason that candymaking is considered one of those semi-advanced cooking topics. If it was easy, we'd probably all overdose on fudge by the time we learned to safely operate a stove.
Best Answer
As you suggest, most nougat recipes require the sugar syrup to be brought to specific temperatures (typically "hard ball" or "soft crack" stage, see for example this page for details of the stages of cooking sugar). By that time, any water that you started out with (in your case in the form of rose water) has boiled off. Simply bringing the mixture up to a boil will not be accurate.
For repeatable results, I suggest you use an instant-read thermometer (or a sugar thermometer) to ensure your sugar syrup is at a consistent temperature. If the nougat does not set, increase the temperature next time.
Regarding your second question: Even in your recipe, you add some of the syrup to the egg whites, to raise their temperature before adding them to the syrup. This is called 'tempering' and is common when using eggs in sweet applications with something hot. In the case of nougat, however, you can probably add the egg whites directly to the syrup, as was pointed out in a comment.