If you are booked on this:

1 A3 610 ATHCDG 835A1105A

2 UA 914 CDGIAD 1145A 250P

You are on an illegal connection and UA is not obligated to reprotect you if you miss it.

It's the next available flight in coach, which might be several days.

Even if you don't want to believe us that it's an illegal connection, think about the logistics: You must first deplane with baby, while carrying baby equipment. There's a chance that the Aegean flight will arrive at a bus gate rather than a jetway, so you wait until the bus is fully loaded, drive around the airport, stop at some door where everyone pushes out to a door and steps to the main level of the airport. Then you look for monitors to check your gate and signs directing you toward the new gate. Along the way, you MUST stand in line for passport control and TSA-style airport security, including inspection of any liquids for the baby since you will be flying a USA carrier to the USA. USA legacy carriers generally have rules that you are required to be on board 30 minutes before scheduled departure time for international flights or you can be denied boarding. That gives you exactly TEN MINUTES to deplane with baby/stuff, get through passport control and security, and find the new gate. It's not going to happen.

That's what I was thinking. How could the airline put together a connection like that if they thought there was a decent chance of not making it in time. And surprisingly they did actually move me to the bulkhead when I called in and mentioned the bassinet. I've read some horror stories online from quite a few people about United and bassinets. The bassinets and bulkhead seats would be reserved, but United never actually booked them or changed the seats on the reservation and then sold them to other people instead. So I was actually quite impressed with them for making a change that I could actually see on my reservation right away.

Can the same flight have two different names? My flights are UA9848 and UA914

Yes. UA9848 is simply the UA codeshare number for Aegean flight A3 610. It is an Aegean plane.

No idea how the UA agent thinks the A3 gate assignment will be near the UA gate at CDG.

UA9XXX is a codeshare for flights operated by a partner, in this case Aegean. Yes, the same flight can be known by different flight numbers and different "names" of the carrier that is selling the ticket. No UA airplanes fly between CDG and ATH. In Star alliance, it would be Aegean (unless there's a stop in Germany, then it would be Lufthansa).

Yep, intraEU or schengen flights and long hauls to the USA operate from different areas at CDG, separated by passport control for instance.

After talking to 2 or 3 more agents, I got one to switch me to the later Paris flight without even a change in fare, so I now have and 1 hour and 35 minutes to make my connection.

My flight back has a layover from San Diego to Chicago with only 40 minutes to spare though. One of the agents insisted the Chicago layover was no issue because I wouldn't have to go through passport control or security again (he was concerned about the Athens/Paris transfer and suggested I either cancel my ticket or pay to change)

Can you please list the flights you are talking about? As there is only one flight from CDG to Washington, are you now going via San Diego and Chicago?

As I listed above, 90 minutes is the minimum valid connection time in CDG so you now have a valid connection.

Based on my reading of her previous posts, she's saying that on the return trip (SAN-ATH), she has a connection time of 40 minutes in Chicago and is asking about that connection time now.

Even if it's legal, it doesn't sound wise when traveling alone with a baby.

I just did a dummy booking and united.com will indeed sell that ticket.

It's also clear that 40 minutes violates any theoretical minimum connection time.

We can debate what UA's responsibilities will be in case of a missed connection. Since it is a single PNR on UA stock, I believe they would be responsible for rebooking and accomodation. BUT you really don't want to be having that argument with them. Particularly since you will be dealing with notoriously awful contract employees at the CDG outstation.

Much better to call back and switch to an earlier flight on either the LX (Swiss) codeshare through ZRH (UA7709-UA937) or one of the LH (Lufthansa) codeshares through FRA (UA9301-UA998 or UA9801-UA8807). Ask for a supervisor if the agent will not make the change. If you can, go for ZRH, it's a better airport for connections than FRA.

Edited to all: Oh well, looks like OP has already rebooked.

Yes, that's the flight I'm referring to. I agree with everyone that the 40 minute Chicago layover is pretty risky too, but I think I'll just stick with that one and hope everything works out. I've checked many other flights with United and other airlines as well, and most of them have their bulkhead seating sold out. I'd rather risk not making the flight than having to book a flight with a basic economy seat and my son sitting on my lap for 9+ hours

But if you don't make that flight, your options may be even worse than just having your son sitting in your lap for 9+ hours, including having to wait a day or two to get home. If ease of travel with your son is a concern, I would try and get that return flight changed too.