A side side note that I have from bitter experience as it sounds like you are essentially doing exactly what I'm doing. Hopefully it might save you some time (or at least you haven't made the mistakes I have!):

For my upcoming honeymoon I booked F outbound to HKG on a 2-4-1 and booked a return leg on CX via BA as there was no return availability on my inbound dates or anywhere nearby (burning half a 2-4-1, boo ). Of course, this was booked about 6 months prior to my wedding and 12 months prior to when we will be taking the trip as redemption seats go fast.

When it came to changing my wife's name from her maiden name to her married name, we thought it might be a short pain but relatively simple as it surely happens all the time.. Changing some other legs on Qantas took me mere minutes and they comped it - this was on the cheapest economy fares they had. Changing our BA leg was a bit more difficult but after emailing copies of passports and wedding certificates we managed the change, although I can't remember if we had to pay a change fee. I think we did.

Where it got REALLY problematic was getting BA to amend a surname on BA paper for CX metal. They already had evidence of her name change but we had to send it several more times. After a while they claimed it was literally impossible for them to change this record and I would have no choice but to cancel the tickets and rebook. If I had done this, the redemption flights would likely be gone and I would have had to pay around £10k to get my wife equivalent seats.

I have been bounced around from department to department and spoke to well over 12 different agents at Exec Club. Most of the time I could only speak to first-line agents who couldn't understand how to make changes and supervisors (who I was told were the only people who could assist in changes via the ticketing department) were rarely available and when they were would parrot the same "solution" to me as they were told ticketing could do nothing.

I reached out to CX myself and was almost immediately connected to a supervisor without asking. CX customer service was frankly wonderful, especially considering I'm not actually technically a customer of theirs. They reassured me that it was totally possible but needed to be done at BA's end as they ticketed me. The supervisor gave me step-by-step Amadeus instructions for BA but also tried to contact them directly without success, while expressing surprise that I was having to contact them myself as BA should have spoken to CX (BA claimed they could only speak to OneWorld who also told them I could only cancel). She provided me with her work email address and a direct phone number so she could assist me further, which she did throughout the process and even called me on occasion to check how things were going.

Back with BA, I was asked to forward these instructions to generic mailboxes no less than three times, all the way coming back and saying it was not possible. Eventually after getting nowhere I went to BA Customer Relations and after some wrangling managed to get in contact with someone who understood that there was an issue and could understand the instructions. I passed on my CX contact's details and eventually they managed to talk and sort it out.

This process from end-to-end took 2.5 months.

To wrap up, if you need to do the same thing:

1. Don't take no for an answer, it's perfectly reasonable ticket change.
2. BA can change partner bookings and there doesn't need to be an explicit agreement between OW partners on this, despite what I was told.
3. Executive Club (at least at my level, GCH or something might get better) cannot help you even though it's related to Avios. Surprising. Go straight to Customer Relations.
4. At least in my experience, knowledge of BA's own systems appears to be woefully inadequate when dealing with any moderately unusual situation.

Sorry that was a long ramble! I'll be writing a complaint letter to BA once I can formulate my complaint in a clear and concise manner. Needless to say, CX have already received a "thank you" letter Wonderful customer service.