Complex sentence: The majority of those who served in the battle had joined the RAF rather than the Dominions' air forces, which were training large numbers of personnel for service in Europe, under the British Commonwealth Air Training Plan (BCATP), when the battle commenced.
Split into simpler sentences: The majority of those who served in the battle had joined the RAF rather than the Dominions' air forces. The latter were training large numbers of personnel for service in Europe, under the British Commonwealth Air Training Plan (BCATP), when the battle commenced.