Could he take the airside bus if he did not have checked bags - i.e., carry-on only?

I'd agree with the sentiments expressed above that 2.5 hours isn't sufficient time to make the transfer when travelling on separate bookings. There are too many factors at play and too many things that could go wrong. I'd be booking an earlier flight or, depending on the schedule, even planning to fly IBZ - LHR the previous day.

Yes, that would save a lot of time, especially if he was able to do OLCI for the AA flight, then he doesn't really need to worry about the check-in cut-off.

2.5 hrs would be fine (IMO) for a T5-T3 connection on separate tickets with hand bags only. Probably even 2 hours, in which case the only real risk is a meaningful delay on the inbound, but you should still be protected onto a later AA flight.

At LHR if you're transferring from T5 to T3 and you're either HBO or you're travelling with bags that are checked through to your final destination (not the case for the OP) then you can remain airside and take the bus from T5 to T3. You would still need to clear security again at T3 but you would not need to go through UK immigration or customs.

If you need to collect bags at T5 then you'd need to enter the UK to do so which means clearing UK immigration, collecting your bags from the baggage hall, and passing through the appropriate customs channel. You'd then need to transfer from T5 to T3. Details on that are given here at the Heathrow Airport website: https://www.heathrow.com/airport-guide/getting-around-heathrow/travel-between-terminals

For travel on the Heathrow Express train you need an Inter-Terminal Transfer ticket or you can use an Oyster Card or Contactless payment card to go through the barriers. Likewise for the tube where you need an Oyster Card or Contactless payment card.

Customs is very quick. I was never stopped by UK Customs.
The problem for you are long UK immigration lines.

OP: so it sounds like you've already bought your flights, have decided to save the money and risk the tight connection, are uninterested in going with carryon luggage only to save time. Have I got that right?

I suppose the best we can do is wish you luck and to come back and let us know how it went, given that you have the information on how it all is supposed to go.

(Personally I'd ditch the checked luggage and travel with carryon only- I find it easy to do a week in carryon, after that, do some laundry- but you do you.)

What's your backup plan if you miss the LHR-LAX flight?

​​​​​​Assuming that 2.5 hours satisfies the MCT for BA T5 to AA T3 at LHR, presumably the OP's back up plan is to go to AA and have AA ensure that the OP is re-routed to LAX, pursuant to AA's policy which provides protection in the case of IRROPS on separate oneworld-AA tickets.

That's certainly possible.

I've got the following booked through Chase Ultimate Rewards:
08/23 - AA LAX-PHX-LHR
09/06 - BA IBZ-LHR
09/06 - AA LHR-LAX

I am still within the 24 hour period for free ticket change/cancellation, and actually just did a rebook as the price went down $40 overnight, which gets me another 24hrs to consider my options.

I'm considering a flight to London City Airport (LCY), that gets in @ 12:10pm. That would give me 5 hours between flights, but I would have to get from LCY to LHR. Any thoughts on if this is worthwhile?

Ordinarily yes, but with Systemwides in the mix here I don't think I can tempt fate on being re-accommodated and have Systemwides clear again (provided they clear the first time)...

Hi OP, I assume you have already bought the tickets.

In which case, you have only 2 options -- go for it and hope for the best, or restrict baggage to carry-on. If carry-on, you should have no trouble with the timing.

So, if you go for it with bags, do what you can to wriggle through quickly.
1/ get from the plane to immigration quickly. If you land at B or C gates (remote concourse) you need to take the shuttle tram. (there is a walkway but maybe that's a step too far for one unfamiliar with T5). Get on the front of the shuttle. Once off the shuttle, take lifts - elevators, not escalators.
2/ in your favour, I would say that mid-afternoon is quiet at T5. Also in your favour, and this is the biggest factor of all, by September you should be able to use the e-gates.
3/ as you exit customs into the public area, turn right for either tube or Heathrow express (HEX). If you have contactless payment cards (separate one for each passenger) then take the tube. Tap in and out, and it won't charge you anything. It has a 10-minute frequency. Otherwise probably best to take HEX (15-minute frequency). You do not need a ticket from T5 to T3 and there are no barriers. HEX is slightly closer to the AA check-in at T3 than the tube.
4/ Once you have reached a check-in desk at T3 before the 1-hour deadline, you should be fine. Remember you can go to the first check-in desks as an EXP. The security questions happen as part of the check-in, and if you have pre-check they don't happen at all. You then have an hour to do T3 security, and get to the gate. You should be able to use fast track security as an EXP. It's plenty of time, but don't dawdle -- the monitors will be saying go-to-gate as soon as you are through security.

Best of luck

That's sort of my point. I know OP can be put on a subsequent flight as if this was a connecting ticket, but having a plan to deal with a super tight connection probably helps.

Are you keen to wrangle a bunch of checked luggage on the Underground or pay a rather large sum of money for a cab (though less than the $1000 ticket differential)? I guess it could work if those are acceptable answers. Personally... I'd ditch the checked luggage.

I was writing the last post as your last 2 posted.

5 hours LCY/LHR will give you more time. LCY is very small and immigration and customs should be quick. It takes nearly 90 minutes using 3 trains to get to LHR T3. But all in all it should get you there to T3 an hour or hour and a half earlier than the LHR flight into T5 would do. And you should then have time to sample any or all of 7 lounges at T3. (AA 1st/club, BA 1st/galleries, CX 1st/business, QF) You should have a thirst after the Dockland rail/tube journey.

Since when is this true?