IME for BA, AA easy, can all be done online without the need for membership of the said airline FF program. Useful post by BA273 as I hadn't previously been aware of the email option for CX.

For LA (and its call centre, or even the first bit of getting through to it, can be a challenge) the toll free # in the UK is 0800 977 6100 (not sure if that's helpful to you). Other numbers are listed on the site. Should you hold premium status don't be afraid to challenge the agent if what you're first offered is a disappointment.

An easier way of tackling this can be to sometimes book flights under the codeshare designator and, depending on which program you're a member of, this can be very beneficial is there's some kind of positive recirpricity, examples of which for QF members would be anything with a BA or AA designator.

Don't underestimate the importance of getting your seating sorted well in advance, especially if flying business.

Have a great trip.