Turn on an 8-quart multi-functional pressure cooker (such as Instant Pot®), add oil, and select Saute function. Add onions and mushrooms to the hot oil; season with salt and pepper. Cook and stir until onions are turning clear and mushrooms begin to soften, 3 to 5 minutes. Add garlic; cook and stir until fragrant, 1 to 2 minutes. Transfer vegetables to a bowl and reserve.
Pour chicken broth into the pot and add Italian seasoning. Break spaghetti noodles in half and add to the pot in 2 layers, with the second layer in a crisscross direction to the first layer.
Season chicken tenderloins with salt and pepper. Lay on top of the spaghetti. Spoon cooked vegetables over top. Cancel Saute function.
Close and lock the lid and seal the vent. Select high pressure according to manufacturer's instructions; set timer for 5 minutes. Allow 10 to 15 minutes for pressure to build.
Release pressure using the natural-release method according to manufacturer's instructions, for 6 minutes, then use the quick-release method to release any remaining pressure. Unlock and remove the lid.
Add cream cheese, spinach, and nutmeg; stir to distribute ingredients.
Replace the lid and allow the residual heat of the pot to melt the cheese and lightly steam the spinach, about 10 minutes. Give the spaghetti and sauce a final stir, then serve with Parmesan cheese.