I'm trying to get NYC-CUN with avios, for last few months no luck on BA website.

you'll probably need to call since this is on AA. find milesaver availability on the AA website and then call BA to see if they can see the flights. If its like it was a few weeks ago, better to use Skype to call the UK number than wait on hold forever on BA's US line.

I've attempted to get through via phone and punted after 30+ minutes on hold every time. BA is so utterly shameless with their disregard for customers it boggles the mind.

Was calling them several times, agent said he can't find availability, although on AA website it was showing plenty.

Do you have an open and funded Iberia Plus account? If so, you might try looking for the same AA domestic flights there.

There are downsides to booking AA with Iberia Plus: only roundtrip redemptions are permitted, and no changes or cancellations are allowed. But on some AA domestic routes (not NYC-ORD-NYC), you'll pay fewer Avios than what BAEC would charge.

I was told they can't do a search but if you give the agent the exact flight numbers and dates s/he should be able to see them. So, for example, AA2681 JFK-CUN on Sunday, January 13, 2019 (milesaaver award for 15,000 miles [economy class] shows as available on AA.com). Some agents are better than others at knowing how to use the system with its current glitches but if it's available as a milesaaver on aa.com, BA should be able to see it and book it.

Unable to use the BA website the past several days to book two one-way AA domestic US flights. I finally called today when I had bills to pay, etc. and could wait. Total wait time was < 30 minutes, agent could not have been friendlier or more helpful, and no additional fees charged (beyond the standard $5.60/ticket fee). Agent said there was no ETA on system being fixed (but they were trying) and wait time last night exceeded 2.5 hours.

While not optimal, I was very pleased with the level of service received once my call was answered. To be clear, the system being broken has to be a headache (and expensive) for BA.

2.5 hours is unacceptable, whether a paying customer, mileage redemption or ticketing inquiry.

Three days later, still am unable to view any domestic availability through the BA website. "There was a problem with your request, please try again later."

Another downside of redeeming with Iberia is that you won't have BA status benefits for seat selection.. unless you happen to also have Iberia status.
I don't think it's possible online to detach your Iberia Plus # and then attach BA number.

It's not possible to replace an IB FFN with a BA (or AA) FFN using the "Manage My Booking" function at finnair.com?

I don't think you can fully detach the IB FFN from an IB award. I've never had success doing so with any award (ie, detaching original FFN when redeeming that program's miles).

In my particular case, the finnair site shows the IB number on the main itinerary.. but when you go to change/update details, it shows the BA FFN.
When using rj.com, it shows both IB and BA FFNs.

I am a lot of avios points and if they wanted to fix it they would. they are in no rush until summer travel is over im sure