The trans-Siberian railway actually splits in the East. One terminus is Vladivostok and there isn't a way out of there on OW. However, it is also possible to take the train to Beijing, and from there you can pick up BA back to London, or get a further train down through China to Hong Kong where you can pick up CX.

Hopefully in a few months CX will be flying to PEK again so you wil be able simply to pick them up there.