Connecting flight in Athens to the Islands

Is the flight from Athens on a separate ticket from the one that gets you TO Athens?

Will Athens be your first stop in the Schengen zone?

Do you have checked luggage?

Depending on the answers to these questions, the 'answers' to your question are 'yes'; 'maybe'; 'doubtful' ( i.e. scary close)

We will arrive from USA to Athens at 0805 so we will have to go through customs. I am purchasing separate tickets to fly to either Naxos or Milos from Athens, the flight to Naxos is at 1005. We will be flying in April 2022, so Im thinking the customs area should not be too crowded. We will have carryon luggage only.

I would feel comfortable with two hours under those circumstances, but if your first flight is delayed, you need to have all knowledge needed for a backup plan, like when the next flight is, etc.

In my experience 2 hours are sufficient time to catch a connecting flight.
Since you are doing carry on you save 15 to 20 minutes picking up luggage so that's really good.

Get off the flight and follow signs for passport control. Depending on the number of flights all arriving at the same time passport control can be as short as 10 minutes. Once it took us 20 minutes as there were lots of people.

After passport control you walk to the exit doors. There is a sign on one door which says nothing to declare. Walk through and you are in the terminal. No customs.
Go up to the next floor and you are in departures.

I am assuming you are flying Aegean/Olympic. If so they allow you to check in 48 hours before the flight so you will have the boarding pass before you get on the transatlantic flight.

Check carefully regulations for the size of your hand luggage. Both Naxos and Milos are serviced by smaller turbo prop aircraft and there is far less overhead compartment size on those aircraft. You may have to check some bags.The good news is there are bag tag machines where you drop off the luggage. Its efficient and take little time.

Next is security. There are lots of screening stations and this process seldom has taken us more than 10 minutes.

You will probably have sufficient time to get your first taste of Greek beer in the departures area.

Just a thought since you do not seem to have made a decision yet, Naxos is much larger and has a larger permanent population so tavernas stores and services will be open. The tourist season on Milos starts much later. I would choose Naxos.
Here is why
Naxos town https://www.flickr.com/photos/stanbr54/albums/72157632094558042
Trip around Naxos https://www.flickr.com/photos/stanbr54/albums/72157634605629689
Naxos sunsets https://www.flickr.com/photos/stanbr54/albums/72157700728046481

Thanks so much for your detailed reply. I have been very hesitant about Milos for that reason and really wanted to go to Naxos. The next flight after the 1005 flight is not until 1730. I really didn’t want to spend all day in the Athens airport if I could help it! I know it could get stressful if my flight gets delayed but I think I am going to chance it!

Danasman, if your flight begins in USA and is nonstop to Athens, rest easy. THose flights are 99.9% on time, and I've done them for years, even when they are as much as 90 minutes delayed taking off from the US airport. Why?? because they have all that looong loong time over the Atlantic to make up the time.