What fare class is your ticket booked in? If it is a flexible fare, and you can cancel with little or no penalty, you could certainly do that and then just book on AA.

Alternatively, I would suggest that you open a BAEC account, and deposit the 24,000+ miles for the LAX-LHR portion in there.

You can then deposit the LHR-DEL miles in your AAdvantage account. Just make sure that you ask the lounge agents in LHR to put your AAdvantage number in your booking for the flight, and make sure your boarding pass on the LHR-DEL portion has your AAdvantage number on it.

I would do this if you can't cancel because:

1. You could build up your AA account with the miles from the LHR-DEL-LHR segments.

2. You will have close to 25,000 BA miles, which is enough for a roundtrip Miles For Upgrade (MFU) redepmtion from World Traveller Plus to Club World between North America and the UK.

3. Buy enough BA miles to bring your account up to 25,000.

4. If you plan on visiting the UK again, or any country in Europe, you can upgrade yourself roundtrip or yourself and a companion one-way from WTP to CW. This time, make sure you leave from Canada (YVR would be most convenient, I assume), in which case you CAN credit the flights to your AAdvantage account. You'll get bonus for WTP, and you'll get to fly business class.