In a food processor, grind the nuts to a paste, drizzling in the olive oil as needed, until it's as smooth as humanly (or mechanically) possible (don't fret if it's a little grainy).
Crumble in the tofu and blend until smooth. Pour in the soy milk & blend until creamy. Add the garlic, salt, lemon juice, nutmeg & black pepper and puree until well-blended.
If you decide to use mushrooms, sauté them in a little olive oil until tender, about 3 minutes, then transfer them to the food processor. Pulse a few times, just enough to chop the mushrooms and blend them into sauce so that all is evenly incorporated.
Pour the alfredo into a saucepan and cook over medium heat, stirring frequently, until just hot; or you can store it in the fridge in a sealed container until ready to use (use within 2 days of preparation).