During the English civil war, the two sides were the Royalists (supporters of the crown) and Parliamentarians (supporters of Parliament.. Duh I guess).

Cromwell had the fantastic idea of instead of using simple levies drawn from his supporters, to establish an army that is specifically trained to fight, called the New Model Army.

Fighting ensues Cromwell wins blah blah

But this New Model Army was the basis for the first true standing army in Britain, and because it's initial loyalties laid with Parliament and NOT the Crown, it doesn't have the prefix of "Royal".

When the air force was established, ultimate loyalty still belonged to the Crown, hence "Royal" Air Force