If a flight delay causes you to miss a later leg, can you ask to be rebooked on one from another airline, to significantly shorten the total delay?

Most probably you will be stuck. However there can be an almost infinite combination of circumstances that would get AA to talk to UA.

These include (but not limited to):

1. The circumstances surrounding the delay. Was this within AA’s control or out of their control?
2. how much you paid for your fare: if you have a full-fare ticket, expect more accommodation,
3. your status with AA: if you are a tiered flyer with AA, they might cut you a break,
4. load on the AA 10pm flight: if this flight is already full, AA might be better disposed to talk to UA or another airline,
5. the uniqueness of your situation: if there are 5 of you misconnecting to the same final destination, you’re unlikely to get a break unless there’s no room for 5 of you on the 10pm,
6. the length of the delay, especially if it entails the airline paying for a hotel somewhere (which does not seem to be your case),
7. where you are stuck. If you are stuck in O’Hare, which is a hub for both UA and AA, some measure of “reciprocity” between airlines can be expected, i.e. the favour UA does to AA is likely to be eventually returned. The same holds if you are stuck in a big airport with lots of flights by lots of airlines (LAX for instance).

I have seen UA help UA customers find flights on other airlines because UA was overwhelmed (in Denver) that day. I’m sure AA has days when their rebooking system ex some airport cannot cope for whatever reason.

However, it is in my experience very rarely the first (or second) option for an airline to rebook with a competitor. Airlines never guarantee schedule, and this mean you should expect lengthy delays in the case of a misconnect.

I’ve always been told the big 3 (AA,DL, UA) will not protect on each-others airline. The billback is a full coach fare. For example, AA places pax on DL. Delta will billback AA for a full coach fare–which is staggering. It is rare- extremely rare. They can do it, but it will cost them money.
Airlines will also let you fly into another city near by..like Milwaukee instead of Chicago for example. Have you tried that?

Or just suck it up like the rest of us do when there is a delay.

Let’s start with if this is technically possible, that is if American has an e-ticket interline agreement with United, or can issue paper tickets on United. Per expertflyer, AA has the ability to do both, so their computer systems could issue a ticket.

The second question is whether or not they would do this. The contract of carriage describes their obligations: https://www.aa.com/i18n/customer-service/support/conditions-of-carriage.jsp

This isn’t an operational change, but if it were and delays your arrival more than 4 hours, you’d be entitled to a refund, but they say that they will try to accommodate you.

For Force Majeure (weather, and other things they can’t control), they say that refund is your only remedy but that they will also try to accommodate you. Note that they consider public health emergencies and labor strikes to be in this category.

For oversold flights, they say that they will "do our best to get those customers on the next possible flight", presumably because they are required to give you cash for the amount of time you are delayed, and putting you on another carrier can limit how much they have to pay.

For other delays that they cause, generally due to mechanical issues, especially an overnight, they promise to cover the cost of a hotel if they can’t get you on a flight or give you a refund.

So just from the contract of carriage, they only promise to put you on the next possible flight if it’s due to involuntary denied boarding.

This information is consistent with what travel agents are allowed to do when there’s a problem caused by IROPS (mechanical, weather, crew timeouts, etc.): https://saleslink.aa.com/en-US/documents/Archives/TravelNotice/Schedule%20Irregularity%20(IROPS).pdf

Back to your question, depending on the reason for the delay, it’s possible that AA will hold the outbound flight so that there’s not a misconnect. If they decide to not hold the flight, then you’re only guaranteed to be booked on another American flight. American can technically put you on the United flight, so now let’s look at the likelihood that they will.

At least as of 2018, American publicly stated that their policy is to only put you on another airline if you have elite status, and only if you’re delayed more than five hours if you aren’t Concierge Key, Executive Platinum or Platinum Pro. Since you would be delayed more than five hours, you have a reasonable chance if you’ve qualified for any status in their frequent flyer program.

If you don’t have this status, the agent would be bending the rules to accommodate you on another airline. Sometimes being very nice to an agent having a bad day can help, and sometimes making the request from an airport lounge can help too, but I wouldn’t count on it.

If you think it’s probable that you’ll misconnect and you don’t have American frequent flyer status, I’d be prepared to be stuck in the connection city until the next flight.

It happens. I wouldn’t count on it, but I have had it happen to me.

I had a flight from Frankfurt, Germany to Chicago in the US. From Chicago, I had a connecting flight to Nebraska.

Due to delays in the flight from Frankfurt to Chicago, I was too late for the connecting flight to Nebraska.

United put me on a Delta flight from Chicago to St. Louis and then to Nebraska. That "dog leg" through St. Louis got me to Nebraska much sooner than waiting for the next United flight from Chicago to Nebraska.

Can happen, does happen.

Despite getting me to my destination on time, the side trip through St. Louis with Delta caused some other problems.

Delta managed to lose a large Pelican case containing some rather expensive equipment. They couldn’t find it. Since the whole point of the trip was the equipment, I couldn’t do the things I had planned for a couple of days – it was a business trip, not vacation.

United eventually found my equipment in a Delta storage room in St. Louis and got it delivered to me.

Even if you manage to talk American into putting you on a flight from another airline, it may not go all that smoothly.

Can you ask them to put you on a different airline? Sure! It’s a free country! You can even ask them to rent you a private plane and fly you to your destination. Doesn’t mean they will though…

So the next obvious question is "Can they do it?", and the general answer is "yes", or more correctly, "it depends". Technically an airline can "endorse" your ticket to another airline, as long as they have a relationship with that airline that allows them to do that. In the example you gave, American Airlines and United Airlines do have such an agreement, so they can do it. However (for example) Southwest Airlines do not have such agreements with other airlines, so neither United or American could move you to a Southwest flight, nor vice versa. There are also additional criteria like the fact that the airline must have space available on their flight (and without getting too technical, this normally needs to be "positive space" seats, which means that one airline can’t overbook on another – so even if the other airline is still selling tickets on a flight your carrier might not be able to move you to that flight), and there needs to be sufficient time for you to get onto the new flight – potentially including your luggage if you have any (especially for an international flight).

Now we get to "Will they do it?", and the answer is once again, "it depends". The airlines all have internal policies around when they can do this, with criteria including things like how long your delay would be, what the cause of the initial delay was (eg, if it was their fault, or something like weather), whether they will have additional costs for keeping you on their flights (eg, paying for a hotel or meals), how full their later flights are, your importance to the airline (frequently flyer status, etc), whether or not you have checked bags that will need to be reunited with you, etc. The airlines don’t publish these procedures, so there’s no real way of knowing exactly which criteria each airlines uses and thus what your chances of them saying "yes" will be.

Finally, we get to "Do they have to do this if you ask?", and then answer for that one is a simple "no". You can always ask, but they can always say "no". Simple as that!

So can you ask? Sure! Should you ask? Why not! Will you be successful? It depends – although realistically in most cases, probably not!

You can certainly try. Chances of success depend a lot on your status with American and fare class of your ticket.