If the issue is that you turn the key and the engine doesn't even try to crank, you likely have an electrical issue somewhere between the ignition switch and the starter. You've replaced the starter recently, so lets assume you can rule that out. My next step would be looking for and replacing the starter relay. Past that the ignition switch. To be 100% sure, if you have access to a lift and can check the voltage at the starter when you turn the key, that will be quite valuable plus will help rule out parts so you're not just chucking money at it.
Also it's worth noting that I'm having a similar situation with my RS4. The longer it sits especially in cold weather, the more times I have to cycle the key before the starter kicks in. For me, I'm almost 100% it's the starter, however if yours was replaced not that long ago, seems odd. For me it continued to get worse until it took sometimes 10 minutes for it to finally fire. If I add a small booster pack it fires up no problem. My thinking is the winding in the starter solenoid are worn and when cold don't make great contact until warmed up.
If you're having an issue where the starter is engaging and the engine is cranking a long time (turning over) before it catches and fires, that's a whole other ball game. Then you can think about fuel supply issues, but if the engine isn't even turning over, it won't be fuel related.
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