That is no different for serverless. You don't magically escape startup times - you need to carefully ensure that cold startup times are low enough, or that you maintain excess capacity to compensate.
The precise extent is different between different platforms depending on overheads, but that just means the point at which you need to trigger scaling is different.
You can find lots of descriptions of approaches people have taken to keep serverless instances running to avoid the cold start delays to work around this... For autoscaling groups you'd instead configure the alarm points used to trigger the scaling accordingly.
Serverless platforms tends to assume the startup will be fast enough to keep the connection open rather than return an error, but that is a load balancer config issue - you can set up retries and wait for other platforms too if it makes sense.
(Though for email this really does not matter - retries for all commonly used mail servers follow some form of exponential backoff up to many hours at least; retries works just fine in practice)